• Euler-Bernoulli
• Friction
• Non-linear dynamics
• wiper blade

The present thesis aims at giving a theoretical contribution for the understanding of some among the main features of wiper system dynamics. More precisely, without searching for the exact solution of the dynamical problem, we aim at investigating some of the main issues by means of different simple models. We have adopted a multi-level strategy. Accordingly, different models have been developed to deal with each level. In this introductory chapter we give a short overview of such models.
The first level consists in the local response of the most deformable parts of the wiper system (i.e., the rubber lip). The results so obtained are used in the second stage of the analysis in which the response of the wiper blade is considered. Finally, the model addressing the response of the whole system can be built on the basis of the results obtained in the previous two stages. We point out that, since each model refers to different parts of the wiper system, each time the word “system” will be referred to the specific part considered.
The first and simplest model addresses the dynamic response of the cross-section of a wiper blade within its own plane. In particular, the deformable rubber lip and the stick-slip motion to which it can undergo are considered. The elasticity of the rubber lip is taken into account in a very simple way by considering the lip as a single degree of freedom rigid block connected to a linear elastic spring (Figure 1). The block is in turn free to slide with friction on a rigid surface that can be either fixed or moving.
We examine in detail the case in which the surface oscillates according to a prescribed sinusoidal law. Since the model is intentionally kept as simple as possible, the equation of motion can be solved analytically, and the influence of the main geometrical and mechanical parameters on the lip motion can be studied as well. Both the sticking phases and the sliding phases are discussed.
The results obtained by means of this simple model are worth of some consideration. Undoubtedly, the main result is that the evolution of the dynamical system can be forecasted by considering three global dimensionless parameters, namely: the ratio between the surface oscillation frequency and the system’s natural frequency, the ratio between the kinetic and the static friction coefficients, and the ratio between the maximum surface oscillation amplitude and the maximum distance of the block from the origin, which corresponds to it being in a state of static equilibrium. As an original result, the system long-term response can be assessed in terms of the afore-mentioned three parameters, thus allowing for a clear distinction between systems that may allow for stick-slip phases or not.
Increasing the accuracy of our description of the vertebra cross-sectional dynamics, a second model considers the rubber lip as a flexible element endowed with infinite degrees of freedom (Figure 2). The mechanical model allows for effectively studying the non-smooth dynamics typical of such systems. In particular, we make use of the well-known Euler-Bernoulli beam theory, and we assume different boundary conditions within each different phase of the lip motion (i.e., sticking or sliding phase). During each sticking phase, static friction conditions are imposed, and the tip of the beam in contact with the surface is considered as hinged. On the other hand, during each sliding phase, kinetic friction conditions are imposed. A constant force oriented in the opposite direction to that of the velocity is applied at the tip of the beam in contact with the surface and substitutes the constraint on the relative displacement (Figure 3).
A first issue that is investigated regards the most effective representation of the solution of the equations of motion. Firstly, the differential equations of motion are solved analytically by expressing the solution as a superposition of modal shapes and static response functions. By this way, we show that some convergence troubles seem to occur, in particular for what concerns the third spatial derivative and the second time derivative. In the problem at hand, characterised by a chaotic behaviour, such issue is of primary importance. In particular, since the switching between sliding and sticking phases is ruled by the value taken by some quantities in the current configuration, a poor estimation of such quantities may lead to large errors in the determination of the trajectory of the system.
An alternative solution strategy offered by the finite difference method is explored. This solution seems to be comparatively more stable and able to provide a more detailed insight into the short-term dynamical phenomena arising in the neighbourhood of the inversion of the direction of motion of the underlying surface. In this regard, the first set of results obtained shows that the distribution of stiffness along the lip heavily affects the duration and sequence of stick-slip phases.
The results obtained by the models developed at the lip level are worth considering also when the issue of building suitable models for the wiper blade is addressed. From the mechanical point of view, a wiper blade can be thought as an elastic curved beam, supported by an elastic foundation, i.e. the rubber lip. The parameters characterizing the lip mechanical response can be deduced from the previous results.
At the blade level two different schematizations are proposed. In the first one the vertebra is considered as a rigid beam resting upon a nonlinear elastic foundation (Figure 4). At any given point along the beam, when the local maximum static friction force is reached the spring begin to slide over the surface. The model is just a first rough approximation of an actual vertebra. Nonetheless, it enables characterizing the different regimes of motion along the length of the beam, and it seems able to give useful indications about the external torque that should be applied to the beam to keep it in motion with an assigned law.
The second model proposed for the wiper blade consists of two elastic flexible beams, which represent the resistant part of the wiper blade (the vertebrae) and the rubber lip, respectively. An elastic interface connects the two elastic beams, while a nonlinear visco-elastic foundation schematizes the contact conditions between the rough surface and the rubber element (Figure 5).
The equations of motion of the two beams are consistently derived following a direct approach and are expressed with respect to a suitable rotating frame. More precisely, although the rotations of the cross sections of the beam and the displacements of its line of axis can be large, the deformations can be considered small with respect to a properly chosen initial configuration, i.e. the configuration in which the wiper blade is at rest and pushed over the screen. The result is the complete set of coupled partial differential equations describing the lateral and the vertical dynamics of the wiper blade.