Grażyna Sypniewska-Kamińska

Interval finite difference method for solving the one-dimensional heat conduction problem with heat sources

The one-dimensional heat conduction equation with the term concerning some heat sources, together with the mixed boundary conditions is considered. Such problems occur in the area of the bioheat transfer and their well-known example is given by the Pennes equation. The paper deals with some interval finite difference method based on the Crank-Nicolson finite difference scheme. In the approach presented, the local truncation error of the conventional method is bounded by some interval values. A method of approximation of such error term intervals is also presented.

Asymptotic Analysis and Limiting Phase Trajectories in the Dynamics of Spring Pendulum

Spring pendulum is a widely discussed two degree-of-freedom (DOF) mechanical systems in numerous references. In this paper the asymptotic approach and limiting phase trajectories (LPT) have been applied to analyze the two DOF mathematical model of a spring pendulum. The LPT and multiple timescale (MTS) methods are effective tools of the investigation of non-linear systems. Some interesting and important aspects of dynamics of the system are discussed. The main attention is focused on the non-steady-state vibrations when the energy is intensively exchanged. Then with increasing values of the selected parameters, a sudden change in the character of vibrations is observed. These phenomena are very well described by the LPT. The method allows to determine the critical values of the parameters responsible for the mentioned transitions. Our analytical studies are verified by numerical calculations.

Decomposition of the Equations of Motion in the Analysis of Dynamics of a 3-DOF Nonideal System

The dynamic response of a nonlinear system with three degrees of freedom, which is excited by nonideal excitation, is investigated. In the considered system the role of a nonideal source is played by a direct current motor, where the central axis of the rotor is not coincident with the axis of rotation. This translation generates a torque whose magnitude depends on the angular velocity. During the system operation a general coordinate assigned to the nonideal source grows rapidly as a result of rotation. We propose the decomposition of the equations of motion in such a way to extract the solution which is directly related to the rotation of an unbalanced rotor. The remaining part of the solution describes pure oscillation depending on the dynamical behaviour of the whole system. The decomposed equations are solved numerically. The influence of selected system parameters on the rotor vibration is examined. The presented approach can be applied to separate vibration and rotation of motions in many other engineering systems.

Plane Motion of a Rigid Body Suspended on Nonlinear Spring-Damper

The paper deals with the analytical investigation of the behaviour of the harmonically excited physical pendulum suspended on the nonlinear spring. The asymptotic method of multiple scales (MS) has been used to derive approximate solutions in the analytical form. The applied approach allows one to perform a qualitative analysis of the behaviour of the system. MS method gives possibility, among others, to recognize resonance conditions which can appear in the system.