Multiple Non-Smooth Events in Multi-Degree-of-Freedom Vibro-Impact Systems

The behaviour of a multi-degree-of-freedom vibro-impact system is studied using a 2 degree-of-freedom impact oscillator as a motivating example. A multi-modal model is used to simulate the behaviour of the system, and examine the complex dynamics which occurs when both degrees of freedom are subjected to a motion limiting constraint. In particular, the chattering and sticking behaviour which occurs for low forcing frequencies is discussed. In this region, a variety of non-smooth events can occur, including newly studied phenomena such as sliding bifurcations. In this paper, the multiple non-smooth events which can occur in the 2 degree-of-freedom system are categorised, and demonstrated using numerical simulations.     

Modal stability of inclined cables subjected to vertical support excitation

In this paper the out-of-plane dynamic stability of inclined cables subjected to in-plane vertical support excitation is investigated. We compute stability boundaries for the out-of-plane modes using rescaling and averaging methods. Our study focuses on the 2:1 internal resonance phenomenon between modes that occurs when the excitation frequency is twice the first out-of-plane natural frequency of the cable. The second in-plane mode is excited directly, while the out-of-plane modes can be excited parametrically. An analytical model is developed in order to study the stability regions in parameter space. In this model we include nonlinear coupling effects with other modes, which have thus far been omitted from previous models of parametric excitation of inclined cables. Our study reflects the importance of such effects. Unstable parameter regions are defined for the selected cable configuration. The validity of the proposed stability model was tested experimentally using a small-scale cable actuator rig. A comparison between experimental and analytical results is presented in which very good agreement with model predictions was obtained.     

Adaptive Control of Nonlinear Dynamical Systems Using a Model Reference Approach

In this paper we consider using a model reference adaptive control approach to control nonlinear systems. We consider the controller design and stability analysis associated with these type of adaptive systems. Then we discuss the use of model reference adaptive control algorithms to control systems which exhibit nonlinear dynamical behaviour using the example of a Duffing oscillator being controlled to follow a linear reference model. For this system we show that if the nonlinearity is small then standard linear model reference control can be applied. A second example, which is often found in synchronization applications, is when the nonlinearities in the plant and reference model are identical. Again we show that linear model reference adaptive control is sufficient to control the system. Finally we consider controlling more general nonlinear systems using adaptive feedback linearization to control scalar nonlinear systems. As an example we use the Lorenz and Chua systems with parameter values such that they both have chaotic dynamics. The Lorenz system is used as a reference model and a single coordinate from the Chua system is controlled to follow one of the Lorenz system coordinates.     

Comparing the direct normal form and multiple scales methods through frequency detuning

Nonlinear Dynamics - Approximate analytical methods, such as the multiple scales (MS) and direct normal form (DNF) techniques, have been used extensively for investigating nonlinear mechanical...     

Generalised modal stability of inclined cables subjected to support excitations

Parametric excitation is of concern for cables such as on cable-stayed bridges, whereby small amplitude end motion can lead to large, potentially damaging, cable vibrations. Previous identification of the stability boundaries for the onset of such vibrations has considered only a single mode of the cable, ignoring non-linear coupling between modes, or has been limited to special cases. Here multiple cable modes in both planes are included, with support excitation close to any natural frequency. Cable inclination, sag, parametric and direct excitation and nonlinearities, including modal coupling, are included. The only significant limitation is that the sag is small. The method of scaling and averaging is used to find the steady-state amplitude of the directly excited mode and, in the presence of this response, to define stability boundaries of other modes excited parametrically or through nonlinear modal coupling. It is found that the directly excited response significantly modifies the stability boundaries compared to previous simplified solutions. The analysis is validated by a series of experimental tests, which also identified another nonlinear mechanism which caused significant cable vibrations at twice the excitation frequency in certain conditions. This new mechanism is explained through a refinement of the analysis.     

Nonlinear dynamic response and modeling of a bi-stable composite plate for applications to adaptive structures

This paper discusses the formulation and validation of a low order model to capture the dynamics of a bi-stable composite plate, focusing on the dynamics around its stable states. More specifically, the model aims to capture the complex nonlinear subharmonic behavior observed in the dynamic response of the plate. A system identification approach is used to derive simplified equations of motion for the system. Experimental frequency response diagrams are obtained to characterize the observed dynamics in the identification process. Simulations using the identified model are presented showing excellent agreement with the experimentally observed behavior. A theoretical validation of the model is carried out studying the stability of the modes where subharmonic response was observed. Stability boundaries were computed using averaging techniques showing good agreement with experimental results.     

Bifurcations of backbone curves for systems of coupled nonlinear two mass oscillator

This paper considers the dynamic response of coupled, forced and lightly damped nonlinear oscillators with two degree-of-freedom. For these systems, backbone curves define the resonant peaks in the frequency–displacement plane and give valuable information on the prediction of the frequency response of the system. Previously, it has been shown that bifurcations can occur in the backbone curves. In this paper, we present an analytical method enabling the identification of the conditions under which such bifurcations occur. The method, based on second-order nonlinear normal forms, is also able to provide information on the nature of the bifurcations and how they affect the characteristics of the response. This approach is applied to a two-degree-of-freedom mass, spring, damper system with cubic hardening springs. We use the second-order normal form method to transform the system coordinates and identify which parameter values will lead to resonant interactions and bifurcations of the backbone curves. Furthermore, the relationship between the backbone curves and the complex dynamics of the forced system is shown.     

Vibration damping in bolted friction beam-columns

In this paper we examine the use of dynamic friction within a bolted structure to improve damping properties of the structure. The structure considered for this paper consists of two steel beam-columns bolted together allowing dynamic friction to occur at the interface. This paper presents an analysis of the behaviour of the structure and the effect of friction on its dynamics. It also presents an analysis of the energy dissipation in the structure by means of friction and the optimization of the bolt tension in order to dissipate the maximum vibration energy. We define analytical expressions for the vibration behaviour before and after slip occurs as well as the condition at which the slip-stick transition occurs. An experiment, in which the measurements of the bolt tension, the slip within the structure and the bending velocity are made, is used to validate the model. The theoretical analysis gave very close agreement with the experimental results and the effective damping of the structure was increased by a factor of approximately 10 through the use of dynamic friction.     

Periodic and chaotic dynamics in an asymmetric elastoplastic oscillator

We consider the dynamics of a harmonically forced oscillator with an asymmetric elastic–perfectly plastic stiffness function. The computed bifurcation diagrams for the oscillator show regions of periodic motion, hysteresis and large regions of chaotic motion. These different regions of dynamical behaviour are plotted in a two-dimensional parameter space consisting of forcing amplitude and forcing frequency. Examples of the chaotic motion encountered are shown using a discontinuity crossing map. Comparisons are made with the symmetric oscillator by computing a typical bifurcation diagram and considering previously published results for the symmetric system. From this we conclude that the asymmetric system is dominated by a large region of chaotic motion whereas in the symmetric oscillator period one motion and coexisting period three motion predominates.