A note on coefficient of restitution models including the effects of impact induced vibration

In this work multi-modal systems subject to impact are considered. Using energy balance techniques for an arbitrary contact interval the effects of modal vibration can be included. The energy balance is used to obtain a relationship between the coefficient of restitution and the modal energy during the contact period. This allows the effects of impact induced vibration to be considered. The subsequent analytical relationships demonstrate that increasing contact duration and excitation of higher modes can reduce the effective value of the coefficient of restitution. It is also shown how this approach can be related to work on energetically consistent impacts

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

Periodic sticking motion in a two-degree of freedom impact oscillator

Periodic sticking motions can occur in vibro-impact systems for certain parameter ranges. When the coefficient of restitution is low (or zero), the range of periodic sticking motions can become large. In this work the dynamics of periodic sticking orbits with both zero and non-zero coefficient of restitution are considered. The dynamics of the periodic orbit is simulated as the forcing frequency of the system is varied. In particular, the loci of Poincaré fixed points in the sticking plane are computed as the forcing frequency of the system is varied. For zero coefficient of restitution, the size of the sticking region for a particular choice of parameters appears to be maximized. We consider this idea by computing the sticking region for zero and non-zero coefficient of restitution values. It has been shown that periodic sticking orbits can bifurcate via the rising/multi-sliding bifurcation. In the final part of this paper, we describe three types of post-bifurcation behavior which occur for the zero coefficient of restitution case. This includes two types of rising bifurcation and a border orbit crossing event

A note on using the collocation method for modelling the dynamics of a flexible continuous beam subject to impacts

The use of non-smooth modelling techniques to model the dynamics of a flexible impacting beam has recently been reported in Ref. [1]. The method used was based on taking a Galerkin approximation [2] of the partial differential equation (PDE) governing the dynamics of the beam away from impact, and coupling this to a non-smooth coefficient of restitution rule to model the impact [3]. In this letter, the advantages and limitations of using a collocation method instead of the Galerkin method combined with a non-smooth impact law are discussed

Understanding the dynamics of multi-degree-of-freedom nonlinear systems using backbone curves

In this paper we will describe how backbone curves can be used to explain complex dynamic phenomena that can occur in coupled multi-degree-of-freedom physical systems. Three examples will be used to demonstrate some key points. We will describe cases when backbone curves can be decoupled. In the case of nonlinear resonance (or modal interaction) we explain how to distinguish how many modes are interacting, their unison and relative phase characteristics. Bifurcation of higher order interaction curves from the lower order curves will also be discussed. Finally we will consider an example based on the transverse vibration of a thin plate with pinned boundary conditions. Both finite element simulations and a low order differential equation model are developed for this system. The results show the importance of the nonlinear coupling terms in replicating the frequency shift phenomena which is known to occur in structures of this type. Despite its much smaller size, the low order model is able to show qualitative agreement with the finite element model. Knowledge of the backbone curve behaviour for this system, is used to explain the forced damped behaviour

A note on modelling multi-degree of freedom vibro-impact systems using coefficient of restitution models

In this work multi-modal systems subject to impact are considered. Using energy balance techniques for an arbitrary contact interval the effects of modal vibration can be included. The energy balance is used to obtain a relationship between the coefficient of restitution and the modal energy during the contact period. This allows the effects of impact induced vibration to be considered. The subsequent analytical relationships demonstrate that increasing contact duration and excitation of higher modes can reduce the effective value of the coefficient of restitution. It is also shown how this approach can be related to work on energetically consistent impacts

Application of nonsmooth modelling techniques to the dynamics of a flexible impacting beam

Non-smooth modelling techniques have been successfully applied to lumped mass-type structures for modelling phenomena such as vibro-impact and friction oscillators. In this paper, the application of these techniques to continuous elements using the example of a cantilever beam is considered. Employing a Galerkin reduction to form an N -degree-of-freedom modal model, a technique for modelling impact phenomena using a non-smooth dynamics approach is demonstrated. Numerical simulations computed using the non-smooth model are compared with experimentally recorded data for a flexible beam constrained to impact on one side. A method for dealing with sticking motions when numerically simulating the beam motion is presented. In addition, choosing the dimension of the model based on power spectra of experimentally recorded time series is discussed

Dynamics of a two degree of freedom vibro-impact system with multiple motion limiting constraints

We consider the dynamics of impact oscillators with multiple degrees of freedom subject to more than one motion limiting constraint or stop. A mathematical formulation for modeling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom system are considered, along with definitions of the impact map for multiply constrained systems. We consider sticking motions that occur when a single mass in the system becomes stuck to an impact stop, and discuss the computational issues related to computing such solutions. Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. Numerical examples of sticking orbits for this system are shown and we discuss identifying the region, S in phase space where these orbits exist. We use bifurcation diagrams to indicate differing regimes of vibro-impacting motion for two different cases; firstly when the stops are both equal and on the same side (i.e. the same sign) and secondly when the stops are unequal and of opposing sign. For these two different constraint configurations we observe qualitatively different dynamical behavior, which is interpreted using impact mappings and two-dimensional parameter space

Design and testing of a frictionless mechanical inerter device using living-hinges

In this paper a novel type of frictionless mechanical inerter device is presented, where instead of gears, motion of the flywheel is achieved using living-hinges. The design is a type of pivoted flywheel inerter inspired in part by the Dynamic Anti-resonant Vibration Isolator (DAVI) concept, which was first developed in the 1960s. Unlike the DAVI, it will be shown that the pivoted flywheel inerter has the advantage of producing balanced forces. Furthermore the use of living-hinges eliminates the need for gears or other frictional elements in the inerter mechanism. To demonstrate the utility of the new concept, a bench-top experiment was performed using a small-scale living-hinge inerter manufactured using polypropylene hinges. By estimating the experimental system parameters, the transmissibility results from the experiment could be compared to a mathematical model. These results showed that the living-hinge inerter provided an isolation effect of at least three orders of magnitude in terms of the maximum amplitude reduction from the uncontrolled system compared to that with the inerter added. Although friction was eliminated, the living-hinges did introduce additional damping, and this was found to correspond to an increase in the equivalent damping ratio for the uncontrolled system of 1.2%. It is shown that the living-hinge inerter developed in this paper fits all of the essential conditions required to be a practical inerter device. Furthermore, as it operates without mechanical friction, or fluid flow, it represents a new paradigm in experimental inerter technology

A review of the mechanical inerter: historical context, physical realisations and nonlinear applications

In this paper, a review of the nonlinear aspects of the mechanical inerter will be presented. The historical context goes back to the development of isolators and absorbers in the first half of the twentieth century. Both mechanical and fluid-based nonlinear inerter devices were developed in the mid- and late twentieth century. However, interest in the inerter really accelerated in the early 2000s following the work of Smith [87], who coined the term ‘inerter’ in the context of a force–current analogy between electrical and mechanical networks. Following the historical context, both fluid and mechanical inerter devices will be reviewed. Then, the application of nonlinear inerter-based isolators and absorbers is discussed. These include different types of nonlinear energy sinks, nonlinear inerter isolators and geometrically nonlinear inerter devices, many relying on concepts such as quasi-zero-stiffness springs. Finally, rocking structures with inerters attached are considered, before conclusions and some future directions for research are presented