Vol. 4, No. 4 (2015): Regular Issue

Several definitions of fractional differences are discussed. Their applications to fractional maps are compared. As an example, the logistic equation of integer order is discretized by these fractional difference methods. The comparative results show that the discrete fractional calculus is an efficient tool and the maps derived in this way have simpler forms but hold rich dynamical behaviors.

The initial-boundary value problems for the local fractional Laplace equation, which arises in fractal electrostatics, are investigated in this article. The non-differentiable solutions with different initial and boundary conditions are obtained by using the local fractional series expansion method.

Vibroimpact system dynamic behaviour is studied by numerical parametric continuation technique combined with shooting and Newton- Raphson’s methods. The technique is adapted to two-body twodegree- of-freedom vibroimpact system under periodic excitation. Impact is simulated by nonlinear contact interaction force based on Hertz’s contact theory. Stability or instability of obtained periodic solutions is determined by monodromy matrix eigenvalues based on Floquet’s theory. Analysis of dynamic behaviour for specific vibroimpact system was performed. The instability zones, different oscillatory regimes and bifurcation points were found. Poincare sections were also constructed.

Identification of the moving forces on a bridge is essential to bridge design and management. A new method for identifying the time varying axle loads is presented in this research using the bending moment influence line. Based on the theorem of modal superposition and taking the damping force into consideration, several modal accelerations can be obtained from measured accelerations of the bridge at several sections. Based on the d’Alembertian theory, the inertia force of the bridge is expressed as a distributed load. Using the bending moment influence lines of the bridge, the moving force contributing to the moment and the inertia force are obtained. The formulation about flexural moment and moving force is obtained correspondingly. An optimization method is used to find the solution of the equations, and then the moving forces can be obtained at any given time. Examples show that the method has a high accuracy in identifying moving forces even though there are more than one time varying forces. By applying the proposed method, information regarding the moving forces can be obtained without solving the dynamic equation, resulting in an efficient model for applications in engineering.

A matrix-based computational scheme is developed based on the Generalized Harmonic Balance method for periodic solutions of nonlinear dynamical systems. The nonlinear external loading is expanded into a Taylor’s series as a function of displacement and velocity, and is then expressed as a combination of Fourier harmonics through the Generalized Harmonic Balance method. Using the Newton-Raphson’s approach, an iteration scheme is formulated to obtain the solution of harmonic coefficients for the displacement. The present scheme is a general purpose realization of the Generalized Harmonic Balance method in the sense that it does not need an analytical Fourier expansion of loadings, and all of the coefficient matrices involved with the scheme are created in a standard way. An example of a periodically forced Duffing oscillator is provided to demonstrate the performance of the present scheme. Numerical solutions of period-1 motion from the present scheme are compared with numerical results given by the Runge-Kutta method. The numerical results agree well with analytical predictions by Luo et al.

We analyze the vibrational resonance in the Duffing oscillator system in the presence of (i) a gamma distributed time-delayed feedback and (ii) integrative time-delayed (uniformly distributed time delays over a finite interval) feedback. Particularly, applying a theoretical procedure we obtain an expression for the response amplitude Q at the low-frequency of the driving biharmonic force. For both double-well potential and single-well potential cases we are able to identify the regions in parameter space where either (i) two resonances, (ii) a single resonance or (iii) no resonance occur. Theoretically predicted values of Q and the values of a control parameter at which resonance occurs are in good agreement with our numerical simulation. The analysis shows a strong influence of both types of time-delayed feedback on vibrational resonance.

A technique to study the critical chatter conditions arising in the high speed milling machining process is proposed, by means of evaluating system eigenvalue stability and eigenvalue sensitivity with respect to system parameters. Starting from the equations of motion of a general machine tool system, a set of linear, ordinary, time dependent parameter, Periodic Delay Differential Equations (PDDEs) may be the result. Three approaches (multi-step, full-discretization and semi-discretization) based on the extended Floquet theory and time discretization techniques are analyzed. The semi-Discretization (SD) method is considered in order to derive an eigenvalue sensitivity formula and to show the limits of this approach. A different approach not based on the Floquet theory is proposed. A set of linear, ordinary, constant parameter, PDDEs is obtained by applying a spectral decomposition and a generalized harmonic balance technique. The stability of the solution is evaluated by solving a non standard eigenvalue problem, making it possible to predict the occurrence of chatter vibration during milling. The proposed approach makes it possible to obtain a straightforward formula for the sensitivity of eigenvalues, and of the critical chatter condition, with respect to the variation of a system parameter. A numerical example is presented in order to test the proposed approach. Finally, strengths and limits of the proposed technique are critically discussed and a comparison with the SD method is carried out.

The main purpose is to develop an analytic approach for investigating the wave front propagation on a surface of a cylinder that can be used a descriptor of a detonation engine. The analysis is based on a weakly nonlinear approximated gas dynamic equations with incorporated approximation of the Korobeinkov’s chemical reaction model that are used to describe the two-dimensional detonation field on a surface of a two-dimensional cylindrical chamber without thickness. We found that the wave fronts can be expressed analytically in explicit form for special classes of flow (e.g. isentropic gas flow). In more general cases, the dynamics of wave fronts can still be determined explicitly provided that the wave front is known at initial time and the exact solution, or its approximation is known a priori, e.g. from experimental or numerical analysis.