Tom L Hill

The use of normal forms for analysing nonlinear mechanical vibrations

A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations.

Out-of-unison resonance in weakly nonlinear coupled oscillators

Resonance is an important phenomenon in vibrating systems and, in systems of nonlinear coupled oscillators, resonant interactions can occur between constituent parts of the system. In this paper, out-of-unison resonance is defined as a solution in which components of the response are 90° out-of-phase, in contrast to the in-unison responses that are normally considered. A well-known physical example of this is whirling, which can occur in a taut cable. Here, we use a normal form technique to obtain time-independent functions known as backbone curves. Considering a model of a cable, this approach is used to identify out-of-unison resonance and it is demonstrated that this corresponds to whirling. We then show how out-of-unison resonance can occur in other two degree-of-freedom nonlinear oscillators. Specifically, an in-line oscillator consisting of two masses connected by nonlinear springs—a type of system where out-of-unison resonance has not previously been identified—is shown to have specific parameter regions where out-of-unison resonance can occur. Finally, we demonstrate how the backbone curve analysis can be used to predict the responses of forced systems.

Identifying the significance of nonlinear normal modes

Nonlinear normal modes (NNMs) are widely used as a tool for understanding the forced responses of nonlinear systems. However, the contemporary definition of an NNM also encompasses a large number of dynamic behaviours which are not observed when a system is forced and damped. As such, only a few NNMs are required to understand the forced dynamics. This paper firstly demonstrates the complexity that may arise from the NNMs of a simple nonlinear system—highlighting the need for a method for identifying the significance of NNMs. An analytical investigation is used, alongside energy arguments, to develop an understanding of the mechanisms that relate the NNMs to the forced responses. This provides insight into which NNMs are pertinent to understanding the forced dynamics, and which may be disregarded. The NNMs are compared with simulated forced responses to verify these findings.

Screw theory based motion analysis for an inchworm-like climbing robot

Robotica / Volume 33 / Issue 08 / October 2015, pp 1704 1717 DOI: 10.1017/S0263574714001003, Published online: 29 April 2014 Link to this article: http://journals.cambridge.org/abstract_S0263574714001003 How to cite this article: Jianjun Yao, Shuang Gao, Guilin Jiang, Thomas L. Hill, Han Yu and Dong Shao (2015). Screw theory based motion analysis for an inchworm-like climbing robot. Robotica, 33, pp 1704-1717 doi:10.1017/S0263574714001003 Request Permissions : Click here

The Significance of Nonlinear Normal Modes for Forced Responses

Nonlinear normal modes (NNMs) describe the unforced and undamped periodic responses of nonlinear systems. NNMs have proven to be a valuable tool, and are widely used, for understanding the underlying behaviour of nonlinear systems. They provide insight into the types of behaviour that may be observed when a system is subjected to forcing and damping, which is ultimately of primary concern in many engineering applications. The definition of an NNM has seen a number of evolutions, and the contemporary definition encompasses all periodic responses of a conservative system. Such a broad definition is essential, as it allows for the wide variety of responses that nonlinear systems may exhibit. However, it may also lead to misleading results, as some of the NNMs of a system may represent behaviour that will only be observed under very specific forcing conditions, which may not be realisable in any practical scenario. In this paper, we investigate how the significance of NNMs may differ and how this significance may be quantified. This is achieved using an energy-based method, and is validated using numerical simulations.

The importance of phase-locking in nonlinear modal interactions

In nonlinear systems the constituent linear modes may interact due to internal resonance. In this paper we classify two distinct classes of modal interactions: phase-locked interactions, in which there is a specific phase between the interacting modes; and phase-unlocked interactions, in which the modes may interact regardless of their phase. This discussion is accompanied by the study of an example structure in which both classes of interaction may be observed. The structure is used to demonstrate the differences between phase-locked and phase-unlocked interactions, both in terms of their individual influence on the response, and in terms of their influence on each other when both classes of interactions are present.

Experimental Identification of a Structure with Internal Resonance

Engineered structures are becoming increasingly lightweight and flexible, and as such more likely to achieve large amplitude and nonlinear vibratory responses. This leads to a demand for new methods and experimental test structures to see how in practice nonlinearity can be handled. In previous work, the authors studied a continuous modal structure with a local nonlinearity. The structure has been designed to have transparent underlying physics, and easily adjustable natural frequencies, and this leads to the ability to investigate an approximately 3:1 internal resonance between the 1st and 2nd modal frequencies. Therefore the structure exhibits complex responses to harmonic excitation, including isolated regions of the frequency response and quasiperiodic behaviour. In the present work we discuss a rapid means of identifying the structure with the minimum requirements of test data and time. A particular aim is to characterise the underlying linear system using data that is strongly influenced by nonlinearity. A harmonic balance procedure is used to identify a nonlinear discrete spring-mass system, that is modally equivalent to the structure under test. It is found that the inclusion of harmonic components in the test data and the presence of internal resonance leads to surprising amounts of information about modes that are not directly excited by the fundamental stepped-sine excitation.

Nonlinear System Identification Through Backbone Curves and Bayesian Inference

Nonlinear structures exhibit complex behaviors that can be predicted and analyzed once a mathematical model of the structure is available. Obtaining such a model is a challenge. Several works in the literature suggest different methods for the identification of nonlinear structures. Some of the methods only address the question of whether the system is linear or not, others are more suitable for localizing the source of nonlinearity in the structure, only a few suggest some quantification of the nonlinear terms. Despite the effort made in this field, there are several limits in the identification methods suggested so far, especially when the identification of a multi-degree of freedom (MDOF) nonlinear structure is required. This work presents a novel method for the identification of nonlinear structures. The method is based on estimating backbone curves and the relation between backbone curves and the response of the system in the frequency domain. Using a Bayesian framework alongside Markov chain Monte Carlo (MCMC) methods, nonlinear model parameters were inferred from the backbone curves of the response and the Second Order Nonlinear Normal Forms which gives a relationship between the model and the backbone curve. The potential advantage of this method is that it is both efficient from a computation and from an experimental point of view.

Towards a Technique for Nonlinear Modal Reduction

In this paper we discuss an analytical method to enable modal reduction of weakly nonlinear systems with multiple degrees-of-freedom. This is achieved through the analysis of backbone curves—the response of the Hamiltonian equivalent of a system—which can help identify internal resonance within systems. An example system, with two interacting modes, is introduced and the method of second-order normal forms is used to describe its backbone curves with simple, analytical expressions. These expressions allow us to highlight which particular interactions are significant, as well as specify the conditions under which they are important. The descriptions of the backbone curves are validated against the results of continuation analysis, and a comparison is also made with the response of the system under various levels of forcing and damping. Finally, we discuss how this technique may be expanded to systems with a greater number of modes.

Numerically Assessing the Relative Significance of Nonlinear Normal Modes to Forced Responses

Nonlinear normal modes, which describe the unforced, undamped responses of nonlinear systems, are often used for understanding the dynamic behaviour of nonlinear structures in engineering. As such, a key property of nonlinear normal modes (NNMs) is that they relate to the dynamic behaviour of the system when forcing and damping are applied. Previous work has shown that an extremely large number of NNMs may be predicted for relatively simple systems; however only a small number of these NNMs have a meaningful influence on the forced and damped dynamics. As such, a method for determining which NNMs are significant (i.e., strongly relate to the forced dynamics) and which are not is crucial for the use of NNMs as a tool for understanding nonlinear systems.