# Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer
## 1 Introduction  
In structural dynamics, linearisation is a common practice in approximating system behaviour. From a physical perspective, the behaviour is well approximated if deformations are small enough to not exceed the material limit of linear proportionality [1; 2] and displacements do not introduce geometrical nonlinearities [3; 4]. Additionally, devices such as rolling bearings [5; 6], hyperelastic springs/beams [7; 8], non-holonomic constraints [9; 10; 11], and mechanisms such as play and hysteresis [1], friction [12], polynomial forms of restoring forces [13; 14], and bistability [15] are usually neglected if their influences on the macroscopic behaviour of the system are marginal. From an analytical perspective, a linear system can be studied with second order linear differential equations, hence the superposition of the free and forced behaviour of the system. Moreover, modal analysis and techniques directly linked with the linearity of the model can be used [16]. The linearised model behaviour allows the assessment of the stability of the equilibrium [17]. The differences between the actual behaviour of a structure and the predicted one are often explained by the departure from linearity. The linearisation of models is affirmed strategy in Model Order Reduction (MOR) methods, developed to reduce the number of Degrees of Freedoms (DoFs) and, therefore, computational cost, with a marginal loss of accuracy. In structural dynamics, two classes of MOR methods can be identified: data based and model based. Data-based MOR methods are based on a database of previous simulations of the full original nonlinear system [18; 19]. Instead, model-based MOR methods are based on thefull model itself and no preliminary simulations are required. Methods of this class aimed to nonlinear systems have been proposed, both retaining the nonlinearities [20; 21; 22] and linearising them [23; 24; 25; 26]. In [27], a novel model-based MOR method called Multi-Phi was introduced, in which the nonlinear system is projected in the configuration space, followed by a separate reduction of each of the linearised systems. Therefore, the nonlinear system is described through a subset of the mode shapes and the static deformed shapes of each of the linearised systems. The nonlinearities are implicit in the differences between mode shapes and static deformed shapes of the various configurations, which are practically depicted by Finite Element models. The reduced system has a lower computational cost, and it is, therefore, useful for simulations, both in time and frequency domain. The Multi-Phi approach has been adopted in describing the effect of localised non-smooth nonlinearities: in [11], a non-holonomic contact constraint is introduced on a clamped-free beam and two linearised configurations of the system are numerically and experimentally identified. The nonlinear displacement of characteristic beam nodes has been monitored by Keyence Laser spot sensors [28], which demonstrated the methodology effectiveness. The cantilever beam is a common case study in several applications in which nonlinearity contributions are analysed for system dynamics modelling. In [29; 30], a multi-layered beam is analysed assessing the role of different layer distributions in correct dynamic behaviour estimation. Instead, an aeroelastic structural coupling of a beam-based antenna component is identified and discussed in [31]. Moreover, beam dynamics are often exploited for energy harvesting applications, as performed in [32]. Without loss of generality, it is also possible to evaluate localised smooth nonlinearities. Hence, in this paper, different localised nonlinearities are introduced on a cantilever beam with permanent magnet interactions or non-holonomic contacts, which can be described by polynomial form of the restoring forces [33; 34]. The magnets introduce nonlinearity in the overall elastic characteristic with softening or hardening effects, but also may influence the electromagnetic properties due to material nonlinearities [35; 36; 37].