The presence of contact and backlash often leads to complex and rich dynamic responses in mechanical structures. This typically occurs in systems such as jointed structures and geared mechanisms, where the inherent complexity leads to responses which are difficult to predict. In particular, identifying models capable of performing reliable predictions is extremely challenging, both for the definition of the equations of motion and for the characterisation of their parameters. In this chapter, we present a systematic approach based on the restoring force surface method for analysing and identifying the localised piecewise nonlinear characteristics in multi-degree-of-freedom systems. Our approach builds on the knowledge of the underlying linear system and outlines a procedure to identify equivalent nonlinear characteristics via a graphic representation of the piecewise restoring forces. The approach is validated against experimental measurements of a test rig representative of a two-degree-of-freedom system with a localised nonsmooth characteristic. The reconstruction of the nonsmooth restoring force, identification of the piecewise characteristics, and the validation of the model against experimental time histories are performed following our systematic procedure, proving the flexibility of this practical approach.

Meta-heuristic optimisation algorithms are high-level procedures designed to discover near-optimal solutions to optimisation problems. These strategies can efficiently explore the design space of the problems; therefore, they perform well even when incomplete and scarce information is available. Such characteristics make them the ideal approach for solving nonlinear parameter identification problems from experimental data. Nonetheless, selecting the meta-heuristic optimisation algorithm remains a challenging task that can dramatically affect the required time, accuracy, and computational burden to solve such identification problems. To this end, we propose investigating how different meta-heuristic optimisation algorithms can influence the identification process of nonlinear parameters in mechanical systems. Two mature meta-heuristic optimisation methods, i.e. particle swarm optimisation (PSO) method and genetic algorithm (GA), are used to identify the nonlinear parameters of an experimental two-degrees-of-freedom system with cubic stiffness. These naturally inspired algorithms are based on the definition of an initial population: this advantageously increases the chances of identifying the global minimum of the optimisation problem as the design space is searched simultaneously in multiple locations. The results show that the PSO method drastically increases the accuracy and robustness of the solution, but it requires a quite expensive computational burden. On the contrary, the GA requires similar computational effort but does not provide accurate solutions.

Lightweight structures, once ubiquitous in specific sectors such as aeronautics and space sectors, in recent years, have increasingly attracted the attention of industries that, historically, have not been particularly concerned with structural weight. Ample examples are provided by the civil and automotive industry, in which the paradigm shift towards lower carbon footprint and sustainability prompted new trends characterised by mass reduction, the use of novel materials, and the accounting for large deformations. However, accurately modelling the dynamic behaviour of such structures requires nonlinear mathematical models, which are not widely used in common industrial practices. Reduced-Order Models (ROMs) have emerged as a popular alternative to computationally expensive Finite Element (FE) models, nonetheless, there is still a need to evaluate their effectiveness in accurately modelling strongly nonlinear behaviours. This study investigates the capacity of multiple-degree-of-freedom (MDOF) ROMs to capture and predict the nonlinear behaviour of lightweight structures subjected to large deformations. A novel identification procedure, built on existing linear and nonlinear identification methods, is used to identify an ROM from numerical and experimental data. Being based on the separation of the linear and nonlinear restoring force contributions, the proposed method can be easily embedded in the current industrial practices for the identification of mechanical systems, paving the way to an integrated usage of linear and nonlinear dynamic models. To validate the identified MDOF-ROM, a lightweight structure composed of lumped masses and nonlinear elastic connections is experimentally studied and the numerical and experimental results are compared at different excitation conditions. We demonstrated that the existence of the Nonlinear Restoring Force (NLRF) surface in a reduced subspace corresponds to the presence of local active nonlinearities in the experimental model. This information permits simplifying the nonlinear restoring force function of the ROM, improving the overall identification process. Finally, we showed that the identified ROM accurately represents the nonlinear dynamic behaviour of the experimental test rig and correctly predicts the passage from high-amplitude response to low-amplitude response (jumps) when different levels of excitation are applied to the system, demonstrating the effectiveness of the proposed procedure.

In the industry field, the increasingly stringent requirements of lightweight structures are exposing the ultimately nonlinear nature of mechanical systems. This is extremely true for systems with moving parts and loose fixtures which show piecewise stiffness behaviours. Nevertheless, the numerical solution of systems with ideal piecewise mathematical characteristics is associated with time-consuming procedures and a high computational burden. Smoothing functions can conveniently simplify the mathematical form of such systems, but little research has been carried out to evaluate their effect on the mechanical response of multi-degree-of-freedom systems. To investigate this problem, a slightly damped mechanical two-degree-of-freedom system with soft piecewise constraints is studied via numerical continuation and numerical integration procedures. Sigmoid functions are adopted to approximate the constraints, and the effect of such approximation is explored by comparing the results of the approximate system with the ones of the ideal piecewise counter-part. The numerical results show that the sigmoid functions can correctly catch the very complex dynamics of the proposed system when both the above-mentioned techniques are adopted. Moreover, a reduction in the computational burden, as well as an increase in numerical robustness, is observed in the approximate case.

Most of the optimisation studies of Vibration Energy Harvesters (VEHs) account for a single output quantity, e.g. frequency bandwidth or maximum power output, but this approach does not necessarily maximise the system efficiency. In those applications where VEHs are suitable sources of energy, to achieve optimal design it is important to consider all these performance indexes simultaneously. This paper proposes a robust and straightforward multi-objective optimisation framework for Vibration Piezoelectric Energy Harvesters (VPEHs), considering simultaneously the most crucial performance indexes, i.e., the maximum power output, efficiency, and frequency bandwidth. For the first time, a rigorous formulation of efficiency for Multi-Degree of Freedom (MDOF) VPEHs is here proposed, representing an extension of previous definitions. This formulation lends itself to the optimisation of FE and MDOF harvesters models. The optimisation procedure is demonstrated using a planar-shape harvester and validated against numerical results. The effects of changing some structural parameters on the harvester performance are investigated via sensitivity analysis. The results show that the proposed methodology can effectively optimise the global performance of the harvester, although this does not correspond to an improvement of every single index. Furthermore, the optimisation of each performance index individually results in a variety of design configurations that greatly differs from one another. It is here demonstrated that the design obtained with the multi-objective function here proposed is similar to the design obtained when optimising the efficiency.