Search

This paper develops the analytical conditions for the existence of periodic solutions of a second-order,time-delayed, discontinuous dynamical system. A sample model consists of two linear delayed sub-systems with a switching boundary. The defined G-functions for the delayed, discontinuous systems are introduced, and sufficient and necessary conditions for a flow crossing, sliding and grazing along the switch boundary are developed for such a delayed, discontinuous system. Furthermore, nine (9) regular basic mappings in phase plane and thirty-three (33) delay-related mappings for the second-order, time-delayed, discontinuous systems are classified. Constraint equations are predicted analytically for two periodic orbits with initial functions provided posteriorly. Finally, three numerical examples are illustrated to verify the existence of generalized slowly oscillating periodic orbits without and with sliding portions. This paper improves and extends motion switchability conditions at the boundary in discontinuous dynamical systems without delay.

In this paper, analytical solutions of periodic motions in a 1-D nonlinear dynamical system are obtained through the generalized harmonic balance method with prescribed-computational accuracy. From this method, the 1-D dynamical system is transformed to a nonlinear dynamical system of coefficients in the Fourier series. The analytical solutions of periodic motions are obtained by equilibriums of the coefficient dynamical systems, and the corresponding stability and bifurcations of periodic motions are completed via the eigenvalue analysis. The frequency-amplitude characteristics of periodic motions are analyzed through the different order harmonic terms in the Fourier series, and the corresponding quantity levels of harmonic amplitudes are determined. From such frequency-amplitude characteristics, the nonlinearity, singularity and complexity of periodic motions in the 1-D nonlinear systems can be discussed. Displacements and trajectories of periodic motions are illustrated for a better understanding of periodic motions in the 1-D nonlinear dynamical systems. From this study, the periodic motions in the 1-dimensional dynamical systems possess similar behaviors of periodic motions in the van der Pol oscillator.

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.

In this paper, analytically predicted are complex periodic motions in the periodically forced, damped, hardening Duffing oscillator through discrete implicit maps of the corresponding differential equations. Bifurcation trees of periodic motions to chaos in such a hardening Duffing oscillator are obtained. The stability and bifurcation analysis of periodic motion in the bifurcation trees is carried out by eigenvalue analysis. The solutions of all discrete nodes of periodic motions are computed by the mapping structures of discrete implicit mapping. The frequency-amplitude characteristics of periodic motions are computed that are based on the discrete Fourier series. Thus, the bifurcation trees of periodic motions are also presented through frequency-amplitude curves. Finally, based on the analytical predictions, the initial conditions of periodic motions are selected, and numerical simulations of periodic motions are carried out for comparison of numerical and analytical predictions. The harmonic amplitude spectrums are also given for the approximate analytical expressions of periodic motions, which can also be used for comparison with experimental measurement. This study will give a better understanding of complex periodic motions in the hardening Duffing oscillator.

In this paper, periodic and chaotic synchronizations between two distinct dynamical systems under specific constraints are investigated from the theory of discontinuous dynamical systems. The analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator were obtained, and the invariant domain of sinusoidal synchronization is achieved. From analytical conditions, the control parameter map is developed. Numerical illustrations for partial and full sinusoidal synchronizations of chaotic and periodic motions of the controlled pendulum with the Duffing oscillator are carried out. This paper presents how to apply the theory of discontinuous dynamical systems to dynamical system synchronization with specific constraints. The function synchronization of two distinct dynamical systems with specific constraints should be identified only by G-functions. The significance of function synchronization of distinct dynamical systems is to make the synchronicity behaviors hidden, which is very useful for telecommunication synchronization and network security.

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).

In this paper, the chaotic synchronization of the Duffing oscillator and controlled pendulum is investigated. From the analytical conditions developed in [1] , the partial and full synchronizations of the controlled pendulum with chaotic motions in the Duffing oscillator are discussed. Compared with the periodic synchronization, in the chaotic synchronization, switching points for appearance and vanishing of the partial synchronization are chaotic. The control parameter map for the synchronization is developed from the analytical conditions, and the partial and full synchronizations are illustrated to show the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum. For a better understanding of synchronization characteristics between two different dynamical systems, effects with other parameters will be discussed later.

In this Letter, analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator are presented. From the analytical conditions, the invariant domain of such sinusoidal synchronization is determined, and the control parameter map of the synchronicity is achieved. Under specific parameters, numerical illustrations of the partial and full sinusoidal synchronizations of the controlled Duffing oscillator with the pendulum are carried out for a better understanding of such synchronization under specific function constraints. The methodology presented in this Letter is applicable to synchronizations with any specific function constraints.

In this paper, the analytical conditions for a periodically forced Duffing oscillator synchronized with a chaotic pendulum are developed through the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domains are developed. For a better understanding of synchronization of two different dynamical systems, the partial and full synchronizations of the Duffing oscillator with the chaotic pendulum are presented for illustrations. The control parameter map is developed from the analytical conditions. Under special parameters, the two systems can be fully and partially synchronized. Since the forced pendulum has librational and rotational chaotic motions, the periodically forced Duffing oscillator can be synchronized only with the librational chaotic motions of the pendulum. It is impossible for the forced Duffing oscillator to be synchronized with the rotational chaotic motions.

In this paper, a fundamental theory for deformable webs not resisting any compressive membrane forces is developed through a direct derivation on the deformed configuration. In order to better describe the deformable webs, the classification of deformable webs is presented. A theory for the non-continuum elastic network webs consisting of many deformable cables is presented from the deformable cable theory, and then a theory for fabric deformable webs without relative sliding is developed as well. Finally, a nonlinear theory for continuous deformable webs is presented on the deformed configuration. The local criteria for the existence of such deformable webs are presented through the definitions, and such criteria are very significant for the wrinkling stability of the deformable webs. A deformable web possessing the local wrinkling is an unsolved problem in numerical computations. The theory for fabric webs with relative motions needs to be further developed. Herein the fundamental theory for deformable webs is presented only, and numerical examples will be presented in sequel. Such a theory of deformable webs can be applied to textile or other soft materials and bio-membranes.

In this paper, a nonlinear theory for a straight rod is presented from the general theory of the three-dimensional deformable-body in the Cartesian coordinate frame. A set of nonlinear strains is presented, and the stretch on central curve exactly satisfies the deformation geometrical relations. The relations between the Euler angles and deformation are given from the curvatures and torsion curvatures of the central curves, which can easily explain the existing theories of rods and beams. Full nonlinear equations of motion for a nonlinear rod are developed via the vector form. Such a treatise is different from the traditional treatises of nonlinear rods, based on the Cosserat’s theory (e.g., Cosserat and Cosserat [1] in 1896) or the Kirchhoff assumptions (e.g., Kirchhoff [18] in 1859; Love [3] in 1944). This paper extends the ideas of Galerkin [4] in 1915. The nonlinear theory of thin rods can reduce to the existing theories for thin rods and beams, such ideas presented in this paper can be applied for development of the nonlinear theory for plates and shells as well.

The parameter study of a machine-tool with interrupted cutting is completed for eccentricity frequency and amplitude. The effects with respect to chip length are also incorporated, such that comparisons of the parameter maps can be accomplished. Specific areas within the parameter maps are studied to explain the complicated motions within. In such a case, the switching characteristics are shown with respect to eccentricity frequency. The complexity of the periodic solution structure is discussed regarding the stability, in relation to the vector fields and mapping quantities. Furthermore, the traditional definition of a stability boundary is extended beyond existing literature. The most useful data is the overlay of the # of mappings and minimum switching force product record. These measures point out the extent and location of complexity in the machine-tool model studied herein.

The analytical conditions for motion switchability on the switching boundary in a periodically forced, discontinuous system are developed through the G-function of the vector fields to the switching boundary. Periodic motions in such a discontinuous dynamical system are discussed by the use of mapping structures. Two periodic motions and the analytical conditions are presented for illustration. Further investigation should be carried out for a better understanding of the vanishing and stability of regular and chaotic motions.

In this paper, sliding and transversal motions on the boundary in the periodically driven, discontinuous dynamical system is investigated. The simple inclined straight line boundary in phase space is considered as a control law for such a dynamical system to switch. The normal vector field for a flow switching on the separation boundary is adopted to develop the analytical conditions, and the corresponding transversality conditions of a flow to the boundary are obtained. The conditions of sliding and grazing flows to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the corresponding local stability and bifurcation analysis of the periodic motion are carried out. Numerical illustrations of periodic motions with and without sliding on the boundary are given. The local stability analysis cannot provide the proper prediction of the sliding and grazing motions in discontinuous dynamical systems. Therefore, the normal vector fields of periodic flows are presented, and the normal vector fields on the switching boundary points give the analytical criteria for sliding and transversality of motions.

This paper presents the switchability of a flow from one domain into another one in the periodically forced, discontinuous dynamical system. The inclined line boundary in phase space is used for the dynamical system to switch. The normal vector field product for flow switching on the separation boundary is introduced. The passability condition of a flow to the separation boundary is achieved through such a normal vector field product, and the sliding and grazing conditions to the separation boundary are presented as well. Using mapping structures, periodic motions in such a discontinuous system are predicted analytically, and the corresponding local stability and bifurcation analysis are carried out. With the analytical conditions of grazing and sliding motions, the parameter maps of specific motions are developed. Illustrations of periodic and chaotic motions are given, and the normal vector fields are presented to show the analytical criteria. This investigation may help one better understand the sliding mode control. The methodology presented in this paper can be applied to discontinuous, nonlinear systems.

In this paper, the switching dynamics of linear oscillators with arbitrary discontinuous forcing are investigated through the concept of switching systems, and such switching systems consist of countable prescribed linear oscillators with different external excitations. The traditional treatments are to smoothen the discontinuity at switching points of two subsystems in a switching system, which can provide an approximate solution only. Therefore, an alternative method is presented to obtain an exact solution of the resultant switching linear system. Under periodic piecewise forcing and random forcing, the corresponding exact solutions and stochastic responses of switching linear systems are developed. For any periodic forcing, the periodic responses and stability of the resultant system composed of multiple linear oscillators in different time intervals are presented. In addition, the resultant switching system consisting of two oscillators are discussed, and the corresponding stability analysis is carried out.

In this paper, periodic motions for a simplified brake system under a periodical excitation are investigated, and the motion switchability on the discontinuous boundary is discussed through the theory of discontinuous dynamical systems. The onset and vanishing of periodic motions are discussed through the bifurcation and grazing analyses. Based on the discontinuous boundary, the switching planes and the basic mappings are introduced, and the mapping structures for periodic motions are developed. From the mapping structures, the periodic motions are analytically predicted and the corresponding local stability and bifurcation analysis is completed. Periodic motions will be illustrated for verification of analytical predictions. In addition, the relative force distributions along the displacement are illustrated for illustrations of the analytical conditions of motion switchability on the discontinuous boundary.

In this paper, a theory for synchronization of multiple dynamical systems under specific constraints is developed from a theory of discontinuous dynamical systems. The concepts on synchronization of two or more dynamical systems to specific constraints are presented. The synchronization, desynchronization and penetration of multiple dynamical systems to multiple specified constraints are discussed, and the necessary and sufficient conditions for such synchronicity are developed. The synchronicity of two dynamical systems to a single specific constraint and to multiple specific constraints is investigated. Finally, the synchronization and the corresponding complexity for multiple slave systems with multiple master systems are discussed briefly. The meaning of synchronization for dynamical systems with constraints is extended as a generalized, universal concept. The theory presented in this paper may be as a universal theory for dynamical systems. The paper provides a theoretic frame work in order to control slave systems which can be synchronized with master systems through specific constraints in a general sense.

The criterion for grazing motions in a friction-induced oscillator with a time-varying transport belt is obtained. The three mappings for such a friction-induced oscillator are introduced for analytical prediction of the grazing motions. The sufficient and necessary conditions of grazing are expressed from mappings. With system parameter variations, the initial and final switching sets of grazing mapping are illustrated. Numerical illustrations of grazing motions are carried out from analytical predictions. This investigation provides a technique for how to determine the grazing on the time-varying boundary in discontinuous dynamical systems.

The G-functions for discontinuous dynamical systems are introduced to investigate singularity in discontinuous dynamical systems. Based on the new G-function, the switchability of a flow from a domain to an adjacent one is discussed. Further, the full and half sink and source, non-passable flows to the separation boundary in discontinuous dynamical systems are discussed. A flow to the separation boundary in a discontinuous dynamical system can be passable or non-passable. Therefore, the switching bifurcations between the passable and non-passable flows are presented. Finally, the first integral quantity increment for discontinuous dynamical systems is given instead of the Melnikov function to develop the iterative mapping relations.

The grazing bifurcation, stick phenomena and periodic motions in a periodically forced, nonlinear friction oscillator are investigated. The nonlinear friction force is approximated by a piecewise linear, kinetic friction model with the static force. The total forces for the input and output flows to the separation boundary are introduced, and the force criteria for the onset and vanishing of stick motions are developed through such input and output flow forces. The periodic motions of such an oscillator are predicted analytically through the corresponding mapping structure. Illustrations of the periodic motions in such a piecewise friction model are given for a better understanding of the stick motion with the static friction. The force responses are presented, which agreed very well with the force criteria. If the fully nonlinear friction force is modeled by several portions of piecewise linear functions, the periodically forced, nonlinear friction oscillator can be predicted more accurately. However, for the fully nonlinear friction force model, only the numerical investigation can be carried out.

In the first part of this work, the local singularity of non-smooth dynamical systems was discussed and the criteria for the grazing bifurcation were presented mathematically. In this part, the fragmentation mechanism of strange attractors in non-smooth dynamical systems is investigated. The periodic motion transition is completed through grazing. The concepts for the initial and final grazing, switching manifolds are introduced for six basic mappings. The fragmentation of strange attractors in non-smooth dynamical systems is described mathematically. The fragmentation mechanism of the strange attractor for such a non-smooth dynamical system is qualitatively discussed. Such a fragmentation of the strange attractor is illustrated numerically. The criteria and topological structures for the fragmentation of the strange attractor need to be further developed as in hyperbolic strange attractors. The fragmentation of the strange attractors extensively exists in non-smooth dynamical systems, which will help us better understand chaotic motions in non-smooth dynamical systems.

A methodology for the local singularity of non-smooth dynamical systems is systematically presented in this paper, and a periodically forced, piecewise linear system is investigated as a sample problem to demonstrate the methodology. The sliding dynamics along the separation boundary are investigated through the differential inclusion theory. For this sample problem, a perturbation method is introduced to determine the singularity of the sliding dynamics on the separation boundary. The criteria for grazing bifurcation are presented mathematically and numerically. The grazing flows are illustrated numerically. This methodology can be very easily applied to predict grazing motions in other non-smooth dynamical systems. The fragmentation of the strange attractors of chaotic motion will be presented in the second part of this work.

This paper systematically presents a theory for n -dimensional nonlinear dynamics on continuous vector fields. In this paper, a different view to look into the fundamental theory in dynamics is presented. The ideas presented herein are less formal and rigorous in an informal and lively manner. The ideas may give some inspirations in the field of nonlinear dynamics. The concepts of local and global flows are introduced to interpret the complexity of flows in nonlinear dynamic systems. Further, the global tangency and transversality of flows to the separatrix surface in nonlinear dynamical systems are discussed, and the corresponding necessary and sufficient conditions for such global tangency and transversality are presented. The e -domains of flows in phase space are introduced from the first integral manifold surface. The domain of chaos in nonlinear dynamic systems is also defined, and such a domain is called a chaotic layer or band. The first integral quantity increment is introduced as an important quantity. Based on different reference surfaces, all possible expressions for the first integral quantity increment are given. The stability of equilibriums and periodic flows in nonlinear dynamical systems are discussed through the first integral quantity increment. Compared to the Lyapunov stability conditions, the weak stability conditions for equilibriums and periodic flows are developed. The criteria for resonances in the stochastic and resonant, chaotic layers are developed via the first integral quantity increment. To discuss the complexity of flows in nonlinear dynamical systems, the first integral manifold surface is used as a reference surface to develop the mapping structures of periodic and chaotic flows. The invariant set fragmentation caused by the grazing bifurcation is discussed. The global grazing bifurcation is a key to determine the global transversality to the separatrix. The local grazing bifurcation on the first integral manifold surface in a single domain without separatrix is a mechanism for the transition from one resonant periodic flow to another one. Such a transition may occur through chaos. The global grazing bifurcation on the separatrix surface may imply global chaos. The complexity of the global chaos is measured by invariant sets on the separatrix surface. The invariant set fragmentation of strange attractors on the separatrix surface is central to investigate the complexity of the global chaotic flows in nonlinear dynamical systems. Finally, the theory developed herein is applied to perturbed nonlinear Hamiltonian systems as an example. The global tangency and tranversality of the perturbed Hamiltonian are presented. The first integral quantity increment (or energy increment) for 2 n -dimensional perturbed nonlinear Hamiltonian systems is developed. Such an energy increment is used to develop the iterative mapping relation for chaos and periodic motions in nonlinear Hamiltonian systems. Especially, the first integral quantity increment (or energy increment) for two-dimensional perturbed nonlinear Hamiltonian systems is derived, and from the energy increment, the Melnikov function is obtained under a certain perturbation approximation. Because of applying the perturbation approximation, the Melnikov function only can be used for a rough estimate of the energy increment. Such a function cannot be used to determine the global tangency and transversality to the separatrix surface. The global tangency and transversality to the separatrix surface only can be determined by the corresponding necessary and sufficient conditions rather than the first integral quantity increment. Using the first integral quantity increment, limit cycles in two-dimensional nonlinear systems is discussed briefly. The first integral quantity of any n -dimensional nonlinear dynamical system is very crucial to investigate the corresponding nonlinear dynamics. The theory presented in this paper needs to be further developed and to be treated more rigorously in mathematics.

In paper, the sliding dynamics on the separation boundary is discussed based on the set-valued vector field theory. From vector fields in the neighborhood of a specific separation boundary, the passability of the flow from the one domain into another one is further discussed. The switching bifurcation conditions from the passable boundary to the non-passable boundary are developed. The sliding flow fragmentation on the separation boundary surface is also presented. The normal vector product field function is introduced to determine the switching bifurcation and sliding fragmentation.

This paper presents some new ideas to understand the strange attractor fragmentation caused by grazing in non-smooth dynamic systems. The sufficient and necessary conditions for grazing bifurcations in non-smooth dynamic systems are presented. The initial sets of grazing mapping are introduced and the corresponding initial grazing manifolds are discussed. The grazing-induced fragmentation of strange attractors of chaotic motions in non-smooth dynamical systems is presented. The mathematical theory for such a fragmentation of strange attractors should be further developed.

The criterion for grazing motions in a dry-friction oscillator is obtained from the local theory of non-smooth dynamical systems on the connectable and accessible domains. The generic mappings for such a dry-friction oscillator are also introduced. The sufficient and necessary conditions for grazing at the final states of mappings are expressed. The initial and final switching sets of grazing mapping, varying with system parameters, are illustrated for the grazing parametric characteristics. The initial and grazing, switching manifolds in the switching sets are defined through grazing mappings. Finally, numerical illustrations of grazing motions are very easily carried out with help of the analytical predictions. This paper provides a comprehensive investigation of grazing motions in the dry-friction oscillator for a better understanding of the grazing mechanism of such a discontinuous system. The investigation based on the local singularity theory is more intuitive and efficient than the discontinuous mapping techniques.

In this paper, the mechanism of chaotic motion in nonlinear Hamiltonian systems is discussed based on the KAM theory and resonance overlap criterion. The internal resonances and the corresponding chaotic motions are determined analytically for weak interactions. A numerical method based on the energy spectrum is presented for prediction of quasi-periodic and chaotic motions in nonlinear Hamiltonian systems. The presented numerical method can be applied to integrable, nonlinear Hamiltonian systems with many degrees of freedom. A 2-DOF integrable, nonlinear Hamiltonian system is investigated as an example for demonstration of the procedure to numerically determine the chaotic motion in nonlinear Hamiltonian systems. Finally, the Poincare mapping surfaces of chaotic motions for such nonlinear Hamiltonian systems are illustrated. The phase planes, displacement surfaces (or potential domains), and the velocity surfaces (or kinetic energy domains) for the chaotic and quasi-periodic motions are illustrated. The analytical estimates of regular and chaotic motions in nonlinear Hamiltonian systems need to be further investigated. The mathematical theory should be developed for a better prediction of chaotic and quasi-periodic motions in nonlinear Hamiltonian systems with many degrees of freedom.

In this paper, the force criteria for stick and non-stick motions in harmonically forced, friction-induced oscillators are developed from the local theory of non-smooth dynamical systems on connectable domains. The periodically driven, linear oscillator with a simple dry friction is considered as a sampled problem to demonstrate the methodology presented in this paper. With appropriate mapping structures, the force criteria give the analytical predictions of the stick and non-stick, periodic motions for such an oscillator. Furthermore, the effects of external excitations and friction forces on the stick and non-stick motions are discussed, and the corresponding regions of specified motions in parameter space are obtained. The sliding and grazing phenomena for this oscillator are also presented in this paper. However, an extensive investigation on sliding and grazing bifurcations should be carried out in sequel. The displacement, velocity and force responses for stick and non-stick, periodic motions are illustrated for a better understanding of the dynamics mechanism of stick and non-stick motions of the dry-friction oscillator. The force criteria and mapping techniques are also applicable to multi-body contact dynamics. The methodology presented in this paper can be applied for numerical predictions of motions in nonlinear, non-smooth dynamical systems.

In this letter, the real and imaginary flows for non-smooth dynamical systems are described, and the δ-layer of the separation boundary is introduced as well. The onset, existence and vanishing of the sink and source flows in the δ-layer are presented. The switching between the two semi-passable flows and the switching between the sink and source flows are investigated through the singular gluing points. Finally, the necessary and sufficient conditions for the onset, vanishing and switching are presented. These conditions provide the criteria to determine the sliding motion on the separatrix, which can be very easily applied to non-smooth dynamical systems.

In this paper, the mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation is developed, and the mapping structures for specified periodic motions are constructed. Based on the mapping structures, an analytical prediction of all possible, stable and unstable periodic motions is given. The symmetry for the stable, asymmetrical, periodic motions of such a system is observed. However, the unstable periodic motions do not have such symmetry. The methodology presented in this paper is applicable to other non-smooth systems such as friction-induced vibration, impact oscillator and power control systems. In addition, the mapping dynamics provides a useful and efficient tool for the co-existence of periodic motions and/or chaotic motions in nonlinear dynamical systems.

In this paper, an idealized, piecewise linear system is presented to model the vibration of gear transmission systems. Periodic motions in a generalized, piecewise linear oscillator with perfectly plastic impacts are predicted analytically. The analytical predictions of periodic motion are based on the mapping structures, and the generic mappings based on the discontinuous boundaries are developed. This method for the analytical prediction of the periodic motions in non-smooth dynamic systems can give all possible periodic motions based on the adequate mapping structures. The stability and bifurcation conditions for specified periodic motions are obtained. The periodic motions and grazing motion are demonstrated. This model is applicable to prediction of periodic motion in nonlinear dynamics of gear transmission systems.

In this paper, a local theory of non-smooth dynamical systems on connectable and accessible sub-domains is developed. The properties for separation boundaries based on the characteristics of flows are determined, and the sliding dynamics on a specified separation boundary is introduced. The local singularity and transversality of a flow on the separation boundary from a domain into its adjacent domains are investigated, and the bouncing and tangency of the flows to the separation boundary for non-smooth dynamical systems are discussed as well. The sufficient and necessary conditions for the local singularity, transversality and bouncing of the flows are developed. These conditions are applicable for determining complicated dynamical behaviors of non-smooth dynamical systems.

The static pressure and surface shear forces on a rotating, umbrella-deformed disk in an open shroud are investigated numerically to develop a proper dynamical model for investigation of the self-excited vibration of the disk. The fluid force distributions along the radial direction of the rotating disk are obtained by use of a two-dimensional numerical model of the axisymmetric deformation of disk. It is observed that the effects of Reynolds number and deformation amplitude are very significant on the fluid forces. Once the rotating disk is deformed, the vortex asymmetry in the airflow patterns appears in the narrow and wide gaps between the rotating disk and the two fixed walls. With increasing deformation amplitude, the asymmetry becomes more obvious. The vortex becomes strong as Reynolds number increases. In addition, it is observed that the transition flow between the laminar and turbulent flows exists in the narrow-gap side. Such a transition flow causes the fluid forces to be non-smooth in the corresponding region, and the force discontinuity will lead to the complicated vibration of rotating disk. From this investigation, the fluid forces for different modes of the rotating disk should be further investigated analytically and numerically for a better understanding of the mechanism of self-excited vibration in hard-disk drive system.

The closed-form solution for rigid-body motions of two-dimensional traveling, sagged cables with a nonlinear geometrical constraint is developed. This closed-form solution shows that the rigid-body motions are always stable even if the translation speed is over the critical speed. With increasing translation speed, the translation motion effects on the cable increase accordingly. When the translation speed is much greater than the critical speed, the vibration effects compared to the translation motion effects can be ignored. Therefore, for an infinite translation speed, the initial and dynamic configurations for the inextensible cable are identical. The dynamic configuration of the inextensible cables is obtained in this paper, which provides a basis to get the dynamic responses of sagged, elastic cables.

The methodology is presented in this paper through investigation of the equilibrium and stability of thin plate axially traveling with high-speed, which is applicable for traveling plate and shell structures. Equilibrium, membrane forces and buckling stability for the steady state motion of axially traveling, thin plates are obtained analytically. The equilibrium solutions and membrane forces for a single mode are discussed as a sample. The transverse displacement amplitude and the buckling stability are presented. The results are reducible to the linear, von Karman and nonlinear membrane theories. Finally, the application range of those analytical results is discussed. Such an investigation provides some useful results for numerical simulations and engineering design. For higher modes, the nonlinear plate theory should be used even if the deflection amplitude is very small compared to plate thickness.

The (M:1)-resonant bands in the left and right potential wells are skew-symmetric, and the (2M:1)-resonant bands of the large orbit motion are symmetric. The analytical conditions for the onset and destruction of a resonant band are developed through the incremental energy approach. The numerical predictions of such onset and destruction are also completed by the energy increment spectrum method. The sub-resonance interaction occurs for strong excitations, which needs to be further investigated. These results are applicable to the small- and large-orbit motions of post-buckled structure under a parametric excitation.

The periodic impact motions of passengers in a vehicle traveling on rough terrain are investigated through a linear model of vehicle and passenger systems. In this mechanical model, the human body is considered as a massless bar with a lumped mass. The linear model assumes that the motion response of vehicle is quite small compared to the passenger's pitch motion since the vehicle chassis has a quite large mass and large moment of inertia compared with each passenger. The period-1 motion pertaining to two impacts, respectively, on two walls during N-periods of the ground motion is predicted analytically and numerically. The stability and bifurcation of such a period-1 motion are determined. The impact motion becomes complicated with increasing ground motion amplitude, but the impact motion reduces with increasing torsional damping. Although the mechanical model used in this paper is an ideal one, the dynamic responses of the human body in vehicle traveling on rough terrain is useful for the rough evaluation of the human body safety in traveling vehicles.

Stability, saddle-node and period-doubling bifurcation conditions for the LR model motion in a horizontal impact oscillator are determined analytically and numerically. The regions for such conditions are developed in parameter space. The stability and bifurcation conditions for all the motions of this impact oscillator depend strongly on the initial impact phase instead of excitation frequency. The chaotic motion of the LR model induced by the period-doubling bifurcation are investigated numerically. The period doubling, periodic motions are illustrated through displacement responses and phase planes. The strange attractors of the chaotic motion induced by period doubling are presented by use of phase planes and the Poincare mapping sections. The analytical and numerical predictions for the stability and bifurcation of the LR motion do not agree well as the non-LR periodic motions coexist with the RL motion, which needs to be further investigated.

The Poincare mapping and the corresponding mapping sections for global motions in a linear system possessing a dead-zone restoring force are introduced through switching planes pertaining to two constraints. The global periodic motions based on the Poincare mapping are determined, and the eigenvalue analysis for the stability and bifurcation of periodic motion is carried out. Global chaos in such a system is investigated numerically from the unstable global periodic motions analytically determined. The bifurcation scenario with varying parameters is presented. The mapping structures of periodic and chaotic motions are discussed. The Poincare mapping sections for global chaos are given for illustration. The grazing phenomenon embedded in chaotic motion is observed in this investigation.

An analytical nonlinear solution for the asymmetric mode vibration of rotating disks is given in this paper through a recently developed, accurate plate theory instead of the von Karman model. The nonlinear solution can reduce to the linear one when nonlinear effects vanish. The symmetrical response is also recovered when the nodal diameter vanishes. The natural frequency varying with rotation speed and deflection amplitude is investigated through a 3.5-inch diameter computer memory disk. From this investigation, it is found that the softening of rotating disks may occur for larger nodal-diameter numbers. The methodology given in this paper can be applied to nonlinear responses in structures such as rotating shafts and traveling plates.

This investigation is to provide a possible prediction for design, manufacturing, testing and industrial applications of a simplified micro-electro-mechanical system (MEMS). The chaotic motion in a certain frequency band of such a MEMS device is investigated, and the corresponding equilibrium, natural frequency and responses are determined. Under alternating current (AC) voltage, the resonant condition for such a system is obtained. It is observed that the lower-order resonant motions can be easily converted to the mechanical force and sensed to the electrical signal. The chaotic motions in the vicinity of a specified resonant separatrix are investigated analytically and numerically. For given voltages, the AC frequency bands are obtained for chaotic motion in the specific resonant layers and resonant motions, and such chaotic motions can be very easily sensed by the output transducer in MEMS.

A theory of stochastic layers is developed for a better understanding of the resonant mechanism of stochastic layers in nonlinear Hamiltonian systems. A criterion based on an accurate whisker map and resonant conditions is developed for prediction of the onset of resonance in the stochastic layer. The onset of a specific primary resonance between the periodic forcing and periodic orbit of the integrable Hamiltonian system in the stochastic layer is predicted analytically. A forced, twin-well Duffing oscillator is investigated as a sample problem for prediction of a specified resonance in the stochastic layer. Verification of the analytical prediction is carried out through a symplectic numerical integration scheme. The analytical and numerical results are in good agreement.

An accurate standard-map approach, instead of the approximate standard-map approach, is developed for the analytical prediction of the onset of a new resonant overlap forming a new resonant-separatrix web in the stochastic layer. This accurate approach applies to the twin-well Duffing oscillator for illustration of the resonant overlap phenomena in stochastic layers. The excitation strengths are obtained analytically for the onset of the specified resonant-separatrix webs in the stochastic layer. This accurate approach gives a very good prediction compared to numerical results for the lower order resonant overlap. When excitation strength is very weak, the accurate and approximate standard-map methods are in good agreement, and the accurate one is applicable to non-linear systems with strong excitations. However, the further improvement should be completed for a more accurate, analytical prediction of the onset of a new resonant overlap in stochastic layers of non-linear Hamiltonian systems with periodical excitations.

An approximate theory of thin plates is developed that is based on an assumed displacement field, the strains described by a Taylor series in the normal distance from the middle surface, the exact strains of the middle surface, and the equations of equilibrium governing the exact configuration of the deformed middle surface. In this theory, the exact geometry of the deformed middle surface is used to derive the strains and equilibrium of plates. This theory reduces to some existing nonlinear theories through imposition of constraints. Application of this theory does not depend on the constitutive law because, the physical deformation measure is used. It can also be applied when the response, loading and geometry of the plate are asymmetric and when the vibration mode number is not small. Predictions of membrane forces in a rectangular plate and of the equilibrium solution in a rotating disk are sample problems.

The energy increment spectrum method is developed for the numerical prediction of the appearance and disappearance of a primary (2 M : 1)-librational or ( M : 1)-rotational resonant separatrix band in a parametrically excited pendulum. The analytical conditions for the presence of such separatrix bands are also obtained. Illustrations of the analytical and numerical results for the appearance and destruction of the resonant bands are given for a comparison.

The analytical conditions for both the onset of a new resonance in stochastic layers and the presence of resonant layers in a periodically driven pendulum are presented. The stochastic layers in the vicinity of the homoclinic orbit and the resonant layers formed near the resonant separatrix are illustrated through the Poincare mapping sections. The similarity between stochastic layers and resonant layers is observed. The mechanism of resonant layers characterized by sub-resonance in nonlinear dynamics can be further investigated through such similarity.

The resonant conditions for traveling waves in rotating disks are derived. The nonlinear resonant spectrum of a rotating disk is computed from the resonant conditions. Such a resonant spectrum is useful for the disk drive industry to determine the range of operational rotation speed. The resonant wave motions for linear and nonlinear, rotating disks are simulated numerically for a 3.5-inch diameter computer memory disk.