150 E. Bonisoli et al. can generate troublesome consequences for the motorbike driveability and stability. Indeed, the motorbikes are intrinsically unstable and thus the primary goal of a motorcycle dynamic design is avoid that this instability comes off during normal functionality. In last years many authors developed rigid bodies mathematical models to analyse the dynamic behaviour of motorcycles, considering only the stiffness and damping of connections and neglecting the flexibility of the main components like chassis and rear swingarm [3–7]. In [8] flexible components are implemented to simulate the frame dynamic behaviour and it comes out that the wheels have a sharp effect on the global dynamics. Multibody approach is used in [9] to study the motorbike drivability and the dynamic response of the rear shock absorber under sudden braking and impacts with curbs. The simulation results of a completely rigid model were compared with another model containing a flexible swingarm, obtained using a floating frame of reference (FFR) approach [10], and highlighting in this last case an unstable behaviour of the motorcycle in the case of subsequent impacts. In [11] Bocciolone et al. characterised the behaviour of the motorbike frame during a high roll angle cornering on high friction passage. It was highlighted that in this condition the torsional stiffness given by the frame flexibility works in series with the front fork: the front fork absorbs only the road irregularity component directed along the shocks axis while the chassis absorbs the other components, working as an equivalent torsion spring. This chassis capability is fundamental to guarantee the adherence in the passages between low and high friction surface. The flexibility of the chassis is also very relevant in comfort studies. Indeed, its high frequencies could be very close to the engine rotation frequencies, causing vibration transmission to the whole structure and a probable fatigue local rupture of a connection between frame and engine. Recently, Lake et al. in [12] give an overview on the effects of flexible components on the global motorcycle dynamics and on the stability properties. They show as the optimal frame stiffness values depend on the purpose of the motorcycle. The low frequency modes are also influenced by the global dynamic properties: the “wobble” mode is stabilised when the frame torsional stiffness is increased, while “weave” mode is stabilised when the bending stiffness is increased. The low frequency wheel dynamics is also important to preserve the tyres integrity; in particular “chattering” phenomena [13] is induced by the violent vibrations of front forks during turning phases. The front wheel can lose contact with the ground because of high variations in the vertical load. The frequency of these vibrations depends on the front unsprung mass and for sports bikes the frequency is about 12–18 Hz. This phenomenon depends on the “stick slip”, generated by the static friction of the sliding parts of the front forks. The chattering is increased by the coupling between the structural vibrating modes of the fork with the in-plane mode of the unsprung mass, and thus it is very important to try to uncouple the two systems to preserve the tyres. In [14] the present authors proposed a mode shape tracing from component-to-system environment using Modal Assurance Criterion. The aim was to follow modifications of components mode shapes in sub-systems of increasing complexity up to the global motorbike frame. In details, four chassis modes, three swingarm modes and four wheel modes were traced from component to assembly, with a particular interest to chassis and swingarm torsion. All the followed component modes had high MAC correlation with respect to the global system, but this does not imply that all of them are influent in the motorbike dynamics and it does not give particular information regarding possible changes in the design to improve system behaviour in working conditions. The paper is organised as follow: in Sect. 14.2 the modal based method is proposed as an extension of DMA. In Sect. 14.3 the method is applied to the case study of motorbike dynamic behaviour with the aim of suggest the components to improve. Finally, some considerations regarding the most influent modes and their impact on motorbike performances are presented. 14.2 Mode Shapes Tracing via Weighted Mode Shapes The proposed method aims to select the most influent components for the dynamic behaviour of a flexible multibody system in specific working conditions. Given a working driving frequency ω to the multibody system, the aim is to move the multibody system resonance frequencies as far as possible with respect to the driving frequency. The requirement of shifting assembly natural frequencies means to re-design and thus to update and improve the flexible components of the systems. The main questions in this case are: (1) which components have a high influence on the assembly mode shapes near the driving frequency? (2) Once selected the most influent components, which mode shapes of these components are mainly participating for the system response at that driving frequency in assembly level? To quickly answer to these questions, a method, intended to obtain a very good dynamic behaviour design of a multibody system, is required. The starting point of the proposed method is the modal analysis of the full assembly, its sub-assemblies, and each single component. Modal analysis is nowadays a standard technique and a very fast tool to perform accurate modal analysis prediction and both Finite Elements Analysis (FEA) commercial software and Experimental Modal Analysis (EMA) tools are widely diffused.
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