For a long time, all stability theorems are concerned with the stability of the zero solution of the differential equations of disturbed motion on the whole region of the neighborhood of the origin. But for various problems of dynamical systems, the stability is actually on partial region. In other words, the traditional mathematical model is unmatched with the dynamical reality and artificially sets too strict demand which is unnecessary. Besides, although the stability for many problems of dynamical systems may not be mathematical asymptotical stability, it is actual asymptotical stability — namely “pragmatical asymptotical stability” which can be introduced by the concept of probability. In order to fill the gap between the traditional mathematical model and dynamical reality of various systems, one pragmatical asymptotical stability theorem on partial region and one pragmatical asymptotical stability theorem on partial region for partial variables are given and applications for gyroscope systems are presented.

In this paper the modified differential quadrature method (MDQM) is proposed for static and vibration analyses of beams. Modified weighting matrices are developed and a new formulation process is presented for incorporating boundary conditions such that the numerical error induced by using the δ-method in the original DQM is reduced. The present method is applied to various beam problems, such as static deflections of Euler beams, buckling loads of columns, and free vibrations of Timoshenko beams. Numerical results of the present method are shown to have excellent accuracy when compared to exact values and are more accurate than those obtained by the original DQM. The accuracy and efficiency of the present method have been demonstrated.

The importance of nonreactive acoustic admittance at a burner exit plane on the flame-driving characteristic was discussed using the measured phase difference between the velocity and pressure oscillations, and the acoustic wave structure inside the duct. Also, the flame-driving characteristics were investigated by flame radiation measurements, and results were compared with those obtained previously from admittance studies. Results show that the comparisons are in good agreement. Also, the flame-driving characteristics are closed related to the acoustic wave structure at the burner exit plane, and could be qualitatively predicted by knowing the imaginary part of nonreactive burner acoustic admittance.

The endochronic cyclic plasticity with finite element analysis (EndoFEM) is employed to simulate plasticity-induced crack closure phenomenon of Al 2024-T3 CCT specimens under maximum cyclic stress of 80MPa and 0.1 stress ratio (R). Various fatigue crack lengths are generated by a rc dominated-node-released strategy. The suitability of element-mesh planning around crack tip is supported by the real simulations in the decreasing tendencies of crack opening load (Pop) with increased distance behind the crack tip, and the enough elements to reflect the reversed plastic responses at minimum load.

EndoFEM results of vertical stress ahead of the crack tip show a typical distribution of small scale yield (SSY) in the realm of fracture mechanics; and Pop/Pmax ratio determined at 1mm behind crack tip is kept constant i.e. Kmax-independent. In these cases, fatigue parameters based on either the far field loading parameter ΔK, the effective ΔK (ΔKeff) with crack closure effect, or the mechanical responses ahead of crack tip (e.g. stress parameter, reversed (plastic) strain at 1mm) are all equivalent and are linearly correlated with the stage II fatigue crack growth (FCP) rate. However, for longer crack length with the ligament bending effect or shorter crack length with the starter notch effect, the Pop/Pmax ratio decreases and changes the SSY stress distribution. This result reduces the usefulness of the above fatigue parameters. As a consequence, a nonlinear correlation of FCP rates with ΔK or ΔKeff are purely empirical. The Kmax-dependent ΔKeff must be considered in the correlation as suggested by the present study of EndoFEM.

An inverse method for identifying the element rigidities of frame structures is developed in this paper. The primary input of this method is the longitudinal strains of the frame elements measured in transient tests. The spectral element method is employed to formulate the equilibrium equation of the frame in the frequency domain. The equilibrium equation is rewritten in terms of the strains of the frame elements. The identification problem is then formulated as an optimization program in which the error norm of the equilibrium equation is minimized. The proposed method is applicable to total structure identification as well as substructure identification. The issue of identifiability is addressed in this study. A numerical example is presented to illustrate the proposed method. The influence of noise is investigated through the example.

Upon encountering a wet or contaminated floor, people often modify their gait and posture to prevent themselves from slipping. This study was conducted to investigate the modification of gait patterns in healthy young adults as they approached and stepped on a slippery floor. Ten females and twelve males were instructed to walk at two different pacer speeds, 90 and 120steps/min, guided by a metronome, on a walkway with two forceplates placed at the center. During the step immediately prior to stepping on a forceplate with or without slippery disturbance, temporo-spatial parameters, selected kinematic parameters, and foot-floor reaction forces were evaluated in each cadence. The results showed that modifications of gait patterns for slip perturbation included shorter step length, increases of flexion angles of hip and knee joints,increases of plantarflexion angles of ankle joint with flattened foot, and decreases of the forward and backward groundreaction forces. However, it was found that such modifications for slip perturbation did not seem to efficiently prevent people from falling.