Mecanismos de acción de los antibióticos

One of the earliest studies on nonlinear dynamics of brake squeal was done by Chargin et al. (1997). A very simplified finite element model was used in their study for predicting the characteristics of squeal noise numerically. Some techniques for optimising the integration time and improving the convergence of analysis were also suggested in (Chargin et al., 1997). Many experimental observations of brakes under squealing condition could be simulated by these numerical results. The transient analysis of the brake model was carried out in MSC/NASTRAN. In those days, Abaqus had not yet developed CEA for systems with frictional contact.

The difficulties of conducting a dynamic transient analysis are fully explained in chapter 5. In brief, the computational cost of this approach has led to a limited usage of this method in practice. As a result, some researchers used different simplification and linearisation techniques in order to include the contributions of nonlinearities in the linearised system. In 1990, Dweib and D‟Souza conducted a comprehensive study on a pin-on-disc system. In the first part (D‟Souza and Dweib, 1990), the experimental setup of their study was described. More interestingly, in the second part (Dweib and D‟Souza, 1990), they presented the results of the linear stability analysis along with limit cycles of nonlinear oscillations. For predicting the limit cycle motion of the system, they employed the method of “describing functions” (DFs). The friction force was defined by a polynomial of order three. This polynomial could represent the regime of the friction force in the static and dynamic states. Employing the method of describing functions could overcome the difficulties of doing the transient analysis.

It is worth mentioning that depending on the form of nonlinearities, the method of describing function can become mathematically arduous. It is fairly straightforward to deal with cubic nonlinearities via DFs. The “harmonic balance method” (HBM) can also be used for this purpose. On the other hand, if a non-smooth nonlinear function is used by these methods several harmonic terms must be included in the analysis, which makes the derivation of the required terms very difficult. As the friction force has a non-smooth behaviour, the implementation of these methods is still not very straightforward. Moreover, the application of these methods for large scale finite element models has only been done for cubic nonlinearities and a simplified brake model (Coudeyras et al. 2009a; 2009b).

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In 2007, Massi et al. employed both complex eigenvalue analysis and dynamic transient analysis for investigating the squeal occurrence of a simple model consisting of a disc, a small friction pad and a supporting beam. The main conclusion was that the onset of instability could cost-effectively be predicted with CEA. CEA overestimated the number of unstable mode, as expected due to the absence of damping. Then, the nonlinear time-domain analysis was carried out in their study to demonstrate the significant role of the contact in stabilising the nonlinear oscillations on a limit cycle. Both CEA and transient analysis were therefore considered essential for predicting the squeal occurrence.

Theoretically if a bifurcating periodic solution (limit cycle) is stable, the bifurcation is called “supercritical”, while an unstable bifurcating periodic solution is called “subcritical”. In 2009, Hochlenert published a paper in which the existence of a subcritical Hopf bifurcation was investigated for the brake squeal problem. Hochlenert conducted this study to justify some observations from physical test rigs, which had not been covered by the existing theories. For example, according to the experimental observations, the frequency of squeal was independent of the rotational speed of the disc. However, squeal could only occur when the disc speed was below a certain level.

It was believed in Hochlenert‟s study that the reason for the afore-mentioned phenomenon is the existence of a subcritical bifurcation. In his study, the centre manifold theory was used for nonlinear stability analysis of a simplified brake model consisting of 12-degrees-of- freedom. The centre manifold theory with the application for friction-induced instability was fully explained in (Sinou et al. 2003; Sinou et al. 2004). The same brake model as the one in Hochlenert‟s study was used in (Von Wagner and Hochlenert, 2011) for making this point that nonlinearities could affect the stability boundary of a system, and therefore the linear stability analysis might be insufficient for predicting unstable modes.

Grange et al. (2009) proposed a method in which the nonlinear dynamic response of a pad- beam squealing contact was replaced with an equivalent linear system. The equivalent response was found by a spectral linearization of the nonlinear system with unilateral contact and friction conditions.

In 2012, Kruse et al. applied the harmonic balance method (HBM) to determine the limit cycle of nonlinear oscillations of a minimal brake model. In their study, the characteristics of joints and connectors in brakes were taken as the major source of nonlinearity. The application of HBM helped identify the critical unstable modes leading to squeal. In the authors‟ view, the focus of brake engineers on uncritical unstable modes would be time- wasting if they were not supposed to generate squeal. Therefore, it was better to identify the squealing via HBM and target them for squeal suppression.

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Vermot des Roches et al. (2013) introduced a model reduction method in order to make transient analysis practical for industrial applications. This method which was called “component mode tuning” was applied on a train brake. Vermot des Roches et al. stated that as the only adjustable parameter for brakes in railway industry was pad, the pad must remain unreduced and the other components could be reduced. In this way, nonlinear simulations could be integrated cost-effectively during the design process.