Personalised profiling to identify clinically relevant changes in tremor due to multiple sclerosis.

BACKGROUND: There is growing interest in sensor-based assessment of upper limb tremor in multiple sclerosis and other movement disorders. However, previously such assessments have not been found to offer any improvement over conventional clinical observation in identifying clinically relevant changes in an individual's tremor symptoms, due to poor test-retest repeatability. METHOD: We hypothesised that this barrier could be overcome by constructing a tremor change metric that is customised to each individual's tremor characteristics, such that random variability can be distinguished from clinically relevant changes in symptoms. In a cohort of 24 people with tremor due to multiple sclerosis, the newly proposed metrics were compared against conventional clinical and sensor-based metrics. Each metric was evaluated based on Spearman rank correlation with two reference metrics extracted from the Fahn-Tolosa-Marin Tremor Rating Scale: a task-based measure of functional disability (FTMTRS B) and the subject's self-assessment of the impact of tremor on their activities of daily living (FTMTRS C). RESULTS: Unlike the conventional sensor-based and clinical metrics, the newly proposed 'change in scale' metrics presented statistically significant correlations with changes in self-assessed impact of tremor (max R2>0.5,p<0.05 after correction for false discovery rate control). They also outperformed all other metrics in terms of correlations with changes in task-based functional performance (R2=0.25 vs. R2=0.15 for conventional clinical observation, both p<0.05). CONCLUSIONS: The proposed metrics achieve an elusive goal of sensor-based tremor assessment: improving on conventional visual observation in terms of sensitivity to change. Further refinement and evaluation of the proposed techniques is required, but our core findings imply that the main barrier to translational impact for this application can be overcome. Sensor-based tremor assessments may improve personalised treatment selection and the efficiency of clinical trials for new treatments by enabling greater standardisation and sensitivity to clinically relevant changes in symptoms.

Numerical continuation and bifurcation analysis in aircraft design: an industrial perspective.

Bifurcation analysis is a powerful method for studying the steady-state nonlinear dynamics of systems. Software tools exist for the numerical continuation of steady-state solutions as parameters of the system are varied. These tools make it possible to generate 'maps of solutions' in an efficient way that provide valuable insight into the overall dynamic behaviour of a system and potentially to influence the design process. While this approach has been employed in the military aircraft control community to understand the effectiveness of controllers, the use of bifurcation analysis in the wider aircraft industry is yet limited. This paper reports progress on how bifurcation analysis can play a role as part of the design process for passenger aircraft.

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.

Indirect reduced-order modelling: using nonlinear manifolds to conserve kinetic energy.

Nonlinear dynamic analysis of complex engineering structures modelled using commercial finite element (FE) software is computationally expensive. Indirect reduced-order modelling strategies alleviate this cost by constructing low-dimensional models using a static solution dataset from the FE model. The applicability of such methods is typically limited to structures in which (a) the main source of nonlinearity is the quasi-static coupling between transverse and in-plane modes (i.e. membrane stretching); and (b) the amount of in-plane displacement is limited. We show that the second requirement arises from the fact that, in existing methods, in-plane kinetic energy is assumed to be negligible. For structures such as thin plates and slender beams with fixed/pinned boundary conditions, this is often reasonable, but in structures with free boundary conditions (e.g. cantilever beams), this assumption is violated. Here, we exploit the concept of nonlinear manifolds to show how the in-plane kinetic energy can be accounted for in the reduced dynamics, without requiring any additional information from the FE model. This new insight enables indirect reduction methods to be applied to a far wider range of structures while maintaining accuracy to higher deflection amplitudes. The accuracy of the proposed method is validated using an FE model of a cantilever beam.

Identifying phase-varying periodic behaviour in conservative nonlinear systems.

Nonlinear normal modes (NNMs) are a widely used tool for studying nonlinear mechanical systems. The most commonly observed NNMs are synchronous (i.e. single-mode, in-phase and anti-phase NNMs). Additionally, asynchronous NNMs in the form of out-of-unison motion, where the underlying linear modes have a phase difference of 90°, have also been observed. This paper extends these concepts to consider general asynchronous NNMs, where the modes exhibit a phase difference that is not necessarily equal to 90°. A single-mass, 2 d.f. model is firstly used to demonstrate that the out-of-unison NNMs evolve to general asynchronous NNMs with the breaking of the geometrically orthogonal structure of the system. Analytical analysis further reveals that, along with the breaking of the orthogonality, the out-of-unison NNM branches evolve into branches which exhibit amplitude-dependent phase relationships. These NNM branches are introduced here and termed phase-varying backbone curves. To explore this further, a model of a cable, with a support near one end, is used to demonstrate the existence of phase-varying backbone curves (and corresponding general asynchronous NNMs) in a common engineering structure.

Conditions for the existence of isolated backbone curves.

Isolated backbone curves represent significant dynamic responses of nonlinear systems; however, as they are disconnected from the primary responses, they are challenging to predict and compute. To explore the conditions for the existence of isolated backbone curves, a generalized two-mode system, which is representative of two extensively studied examples, is used. A symmetric two-mass oscillator is initially studied and, as has been previously observed, this exhibits a perfect bifurcation between its backbone curves. As this symmetry is broken, the bifurcation splits to form an isolated backbone curve. Here, it is demonstrated that this perfect bifurcation, indicative of a symmetric structure, may be preserved when the symmetry is broken under certain conditions; these are derived analytically. With the symmetry broken, the second example-a single-mode nonlinear structure with a nonlinear tuned mass damper-is considered. The evolution of the system's backbone curves is investigated in nonlinear parameter space. It is found that this space can be divided into several regions, within which the backbone curves share similar topological features, defining the conditions for the existence of isolated backbone curves. This allows these features to be more easily accounted for, or eliminated, when designing nonlinear systems.

A methodology for identifying optimum vibration absorbers with a reaction mass.

Tuned mass dampers (TMDs), in which a reaction mass is attached to a structural system via a spring-parallel-damper connection, are commonly used in a wide range of applications to suppress deleterious vibrations. Recently, a mass-included absorber layout with an inerter element, termed the tuned mass damper inerter (TMDI), was introduced, showing significant performance benefits on vibration suppression. However, there are countless mass-included absorber layouts with springs, dampers and inerters, which could potentially provide more preferred dynamic properties. Currently, because there is no systematic methodology for accessing them, only an extremely limited number of mass-included absorber layouts have been investigated. This paper proposes an approach to identify optimum vibration absorbers with a reaction mass. Using this approach, a full class of absorber layouts with a reaction mass and a pre-determined number of inerters, dampers and springs connected in series and parallel, can be systematically investigated using generic Immittance-Function-Networks. The advan- tages of the proposed approach are demonstrated via a 3 d.f. structure example.

Passive vibration control: a structure-immittance approach.

Linear passive vibration absorbers, such as tuned mass dampers, often contain springs, dampers and masses, although recently there has been a growing trend to employ or supplement the mass elements with inerters. When considering possible configurations with these elements broadly, two approaches are normally used: one structure-based and one immittance-based. Both approaches have their advantages and disadvantages. In this paper, a new approach is proposed: the structure-immittance approach. Using this approach, a full set of possible series-parallel networks with predetermined numbers of each element type can be represented by structural immittances, obtained via a proposed general formulation process. Using the structural immittances, both the ability to investigate a class of absorber possibilities together (advantage of the immittance-based approach), and the ability to control the complexity, topology and element values in resulting absorber configurations (advantages of the structure-based approach) are provided at the same time. The advantages of the proposed approach are demonstrated through two case studies on building vibration suppression and automotive suspension design, respectively.

A bifurcation study to guide the design of a landing gear with a combined uplock/downlock mechanism.

This paper discusses the insights that a bifurcation analysis can provide when designing mechanisms. A model, in the form of a set of coupled steady-state equations, can be derived to describe the mechanism. Solutions to this model can be traced through the mechanism's state versus parameter space via numerical continuation, under the simultaneous variation of one or more parameters. With this approach, crucial features in the response surface, such as bifurcation points, can be identified. By numerically continuing these points in the appropriate parameter space, the resulting bifurcation diagram can be used to guide parameter selection and optimization. In this paper, we demonstrate the potential of this technique by considering an aircraft nose landing gear, with a novel locking strategy that uses a combined uplock/downlock mechanism. The landing gear is locked when in the retracted or deployed states. Transitions between these locked states and the unlocked state (where the landing gear is a mechanism) are shown to depend upon the positions of two fold point bifurcations. By performing a two-parameter continuation, the critical points are traced to identify operational boundaries. Following the variation of the fold points through parameter space, a minimum spring stiffness is identified that enables the landing gear to be locked in the retracted state. The bifurcation analysis also shows that the unlocking of a retracted landing gear should use an unlock force measure, rather than a position indicator, to de-couple the effects of the retraction and locking actuators. Overall, the study demonstrates that bifurcation analysis can enhance the understanding of the influence of design choices over a wide operating range where nonlinearity is significant.

Inverse dynamics modelling of upper-limb tremor, with cross-correlation analysis.

A method to characterise upper-limb tremor using inverse dynamics modelling in combination with cross-correlation analyses is presented. A 15 degree-of-freedom inverse dynamics model is used to estimate the joint torques required to produce the measured limb motion, given a set of estimated inertial properties for the body segments. The magnitudes of the estimated torques are useful when assessing patients or evaluating possible intervention methods. The cross-correlation of the estimated joint torques is proposed to gain insight into how tremor in one limb segment interacts with tremor in another. The method is demonstrated using data from a single patient presenting intention tremor because of multiple sclerosis. It is shown that the inertial properties of the body segments can be estimated with sufficient accuracy using only the patient's height and weight as a priori knowledge, which ensures the method's practicality and transferability to clinical use. By providing a more detailed, objective characterisation of patient-specific tremor properties, the method is expected to improve the selection, design and assessment of treatment options on an individual basis.

Effects of experimental variables on the nonlinear harmonic generation technique.

Harmonic generation is a promising technique for measuring small changes in the microstructure of components. Its extreme sensitivity is a benefit for detection, but results in a high degree of variability in any measurements taken. This paper characterizes the effects of experimental variables throughout the measurement signal path, establishing their relative importance and making suggestions for the best way to take measurements using harmonic generation. A model is used to predict the harmonic amplitude as a function of position and thereby account for alignment inaccuracy, explaining the effects of this key experimental variable. Finally, the potential effect of all of the variables on damage detection is discussed.

Estimation of upper-limb orientation based on accelerometer and gyroscope measurements.

A solution is proposed to the estimation of upper-limb orientation using miniature accelerometers and gyroscopes. This type of measurement device has many different possible applications, ranging from clinical use with patients presenting a number of conditions such as upper motor neuron syndrome and pathologies that give rise to loss of dexterity, to competitive sports training and virtual reality. Here we focus on a design that minimizes the number of sensors whilst delivering estimates of known accuracy over a defined frequency range. Minimizing the sensor count can make the measurement system less obtrusive, as well as minimising cost and reducing the required bandwidth if using a wireless solution. Accurate measurement of movement amplitude up to 15 Hz is required in our immediate application, namely to quantify tremor in multiple sclerosis patients. The drive for low numbers of sensors and good accuracy at higher frequencies leads to a novel design based on composite filters. The simple estimator structure also gives good insight into the fundamental accuracy limitations based on the sensors chosen. This paper defines the underlying mathematics, and quantifies performance for an estimator for shoulder, upper arm, lower arm and hand orientations. Good estimation accuracy up to 15 Hz is indicated, and this with a reduced total sensor count of 18 compared to 24 that would be required for more conventional estimator architectures.