van de Elb Edward Vorst

Steady-State Behaviour of Flexible Rotordynamic Systems with Oil Journal Bearings

This paper deals with the long term behaviour of flexible rotor systems, which are supported by nonlinear bearings. A system consisting of a rotor and a shaft which is supported by one oil journal bearing is investigated numerically. The shaft is modelled using finite elements and reduced using a component mode synthesis method. The bearings are modelled using the finite-length bearing theory. Branches of periodic solutions are calculated for three models of the system with an unbalance at the rotor. Also self-excited oscillations are calculated for the three models if no mass unbalance is present. The results show that a mass unbalance can stabilize rotor oscillations.

Experimental verification of the steady-state behavior of a beam system with discontinuous support

This article deals with the experimental verification of the long-term behavior of a periodically excited linear beam supported by a one-sided spring. Numerical analysis of the beam showed subharmonic, quasi-periodic, and chaotic behavior. Further, three different routes leading to chaos were found. Because of the relative simplicity of the beam system and the variety of calculated nonlinear phenomena, an experimental setup is made of this beam system to verify the numerical results. The experimental results correspond very well with the numerical results as far as the subharmonic behavior is concerned. Measured chaotic behavior is proved to be chaotic by calculating Lyapunov exponents of experimental data.

Vibration Control of Periodically Excited Nonlinear Dynamic Multi-dof Systems

For nonlinear mechanical systems, which have stable subharmonic resonance peaks and one or more coexisting unstable harmonic solutions, a large reduction of maximum subharmonic, quasi-periodic, or chaotic displacement can be established if the coexisting unstable harmonic solution could be made stable. The control effort to obtain this goal can be very small in that case. In this article, a method for controlling nonlinear multi-degree-of-freedom (multi-dof) systems to unstable periodic solutions is developed. This is established by putting a single control force somewhere on the system. Because the selected control method uses the full state of the system and because only measured displacements and accelerations of a very limited number of dofs are assumed to be available, a reconstruction method has to be used for estimating the full state on-line. Simulations are done using a beam system supported by a one-sided spring that is control led to the unstable harmonic solution. The robustness of the method with respect to model errors, system disturbance, and measurement errors is examined. Further, the performance of the method in case of a varying excitation frequency during the control is investigated.

Long-term dynamics of non-linear MDOF engineering systems

In this paper it is shown how the finite element technique has been integrated with numerical tools for the analysis of non-linear dynamical systems to obtain a tool for efficient dynamic analysis of multi-degree-of-freedom mechanical systems with local non-linearities. The influence of hard discontinuities like elastic stops can be taken into account. The developed methodology is applied to a number of mechanical engineering systems, i.e. a beam system with various discontinuous supports, a rotor model with a non-linear fluid-film bearing and a model of a portable CD player. It is shown that generally models with more than one degree of freedom are required for a sufficiently accurate representation of the system behaviour.

Dynamics of a multi-DOF beam system with discontinuous support

This paper deals with the long term behaviour of periodically excited mechanical systems consisting of linear components and local nonlinearities. The particular system investigated is a 2D pinned-pinned beam, which halfway its length is supported by a one-sided spring and excited by a periodic transversal force. The linear part of this system is modelled by means of the finite element method and subse1uently reduced using a Component Mode Synthesis method. Periodic solutions are computed by solving a two-point boundary value problem using finite differences or, alternatively, by using the shooting method. Branches of periodic solutions are followed at a changing design variable by applying a path following technique. Floquet multipliers are calculated to determine the local stability of these solutions and to identify local bifurcation points. Also stable and unstable manifolds are calculated. The long term behaviour is also investigated by means of standard numerical time integration, in particular for determining chaotic motions. In addition, the Cell Mapping technique is applied to identify periodic and chaotic solutions and their basins of attraction. An extension of the existing cell mapping methods enables to investigate systems with many degress of freedom. By means of the above methods very rich complex dynamic behaviour is demonstrated for the beam system with one-sided spring support. This behaviour is confirmed by experimental results.

Determination of global stability of steady-state solutions of a beam system with discontinuous support using manifolds∗

This paper deals with the global stability of the long term dynamics of nonlinear mechanical systems under periodic excitation. Generally, the boundaries of the basins of attraction are formed by the stable manifolds of unstable periodic solutions. These stable manifolds are the set of initial conditions of trajectories which approach an unstable periodic saddle solution. Because these are the only trajectories which do not approach an attractor, in general the stable manifolds are the boundaries of the basins of attraction. In this paper manifolds are calculated of a beam system supported by a one-sided spring in order to identify the global stability of the coexisting attractors. The numerical results are compared with experimental results.

Manifolds of Nonlinear Dynamic Single-DOF Systems

This paper deals with the long term behaviour (attractors) of nonlinear dynamic single degree of freedom (DOF) systems, excited by a periodic external load. Different attractors can exist for one set of system-parameters. The set of initial conditions of trajectories which approach one attractor is called the basin of attraction of the attractor. The boundaries of the basins of attraction are formed by the stable manifolds of unstable periodic solutions. These stable manifolds are the set of initial conditions of trajectories which approach an unstable period solution (saddle). Because these are the only trajectories which do not approach an attractor, in general the stable manifolds are the boundaries of the basins of attraction. When stable and unstable manifolds intersect, a chaotic attractor or fractal boundaries of basins of attraction are created. These phenomena are demonstrated by calculating the manifolds of two single-DOF systems, one with a cubic stiffening spring and one with an one-sided spring.

Vibration Control of a Nonlinear Beam System

In many engineering applications high amplitude vibrations are undesirable because they may cause wear and damage and may lead to high levels of noise. In nonlinear dynamic systems the steady-state response often exhibits certain frequency ranges where two or more solutions of the system equations coexist. Our objective is to reduce the amplitude of the response by controlling the system into its natural solution with lower amplitude.