Abstract Nonlinear Stochastic Analysis of Motorcycle Dynamics by Luis Alejandro Robledo Ricardo Off-road and racing motorcycles require a particular setup of the suspension to improve the comfort and the safety of the rider. Further, due to ground unevenness, off- road motorcycle suspensions usually experience extreme and erratic excursions in performing their function. In this regard, the adoption of nonlinear devices, such as progressive springs and hydro pneumatic shock absorbers, can help limiting both the acceleration experienced by the sprung mass and the excursions of the suspensions. For dynamic analysis purposes, this option involves the solution of the nonlinear differential equations that govern the motion of the motorcycle, which is excited by the stochastic road ground profile. In this study a 4-degree-of-freedom (4-DOF) nonlinear motorcycle model is considered. The model involves suspension elements with asymmetric behaviour. Further, it is assumed that the motorcycle is exposed to loading of a stochastic nature as it moves with a specified speed over a road profile defined by a particular power spectrum. It is shown that a meaningful analysis of the motorcycle response can be conducted by using the technique of statistical linearization. The validity of the proposed approach is established by comparison with results from pertinent Monte Carlo studies. In this context the applicability of auto-regressive (AR) filters for efficient implementation of the Monte Carlo simulation is pointed out. The advantages of these methods for the synthesis of excitation signals from a given power spectrum, are shown by comparison with other methods. It is shown that the statistical linearization method allows the analysis of multi-degree-of- freedom (M-DOF) systems that present strong nonlinearities, exceeding other nonlinear analysis methods in both accuracy and applicability. It is expected that the proposed approaches can be used for a variety of parameter/ride quality studies and as preliminary design tool by the motorcycle industry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͌3ʚʛ ++( ! +) GGGGGGGG GGGGGGGGGGGGGGGGGGGGG 6 ," ) * "+-+)! +". "! . (/01 GGGGGGGGGGGGGGGGGGGGGGGGGG & G(" ( +) GGGGGGGGGG G $ 3 * +) G-+)! !" GGGG G % G-+)!( -" GGGGG GGGGGGG 4 G-+)!() ++-" GGGGGGGG GGG & 5 )"! )+ ( ! GGGGGGGGG $ 6 7((! ! GGGGGGGG GG G 5 )"! )+ (>9 !" GGGG GGG & $ , +( ! ((7 )! G GGG $4 $ G-+) ( GGGGGGGGGGG GG $6 $ G-+) ( GGGGGGGGGG GG G $6 $ & , +) +-+)!( +( - ++/! G % $ $ G-+) ( . GGGGGGGGGGG %& $ % )(+-+)!(* -(("! %% 611 GGGGGGG G .. GGGG GG . GGGGGGGGGGGGG G G . !" #$ % G GGGGG & !! # $ % GGGGGGGG + !& '()* G ) !+ # $ '()* GGGGGG GGGGGGGGGGGGGGGGGGGGG !) , - GGGGGG GGGGG ! , GGGGGGGGGGGG G !. '()* G GG ! # $ '()* GGGGGGGGGGGGG GGGGGGGGGGGGGG ! , GG GGGGGGGGGG " ! G GGGGGGGGGGGG ! . ! / , GGG GGGGGGGG . & 0 , , GGGG GGGGGGGGGGGG . + 1, % $ , G G . ) ' GGGGGGGGGGGGG GG " . 2$R % GG GGGGGGGGGGG + . )$42 5 GGGGGGGGGGGG G GGGG ) . )$42 5 % GGGGGGG G !"" . 'R )$4 % !". 6111 . 'R )$4 % GGGGGGG GGGGGGGGGGGGGG GGGGGGG !" . 'R )$4 % GGGGGGG GGGGGGGGGGGGGG GGGGGGG !" . 'R )$4 % GGGGGGG GGGGGGGGGGGGGG GGGGGGG !" . GGGGGGGGGGGGGG !" . GG GGGGGGGGGGG !" . !" ' ͔- GGG GGGGGGGGGGGGG !!" . !" ' ͔! GGGGGGG GGGGGGGGG !!" . !" ' ͔+- GGGG GGGGGGGGGGG !!" . !" ' ͔+! G GGGGGGGGGGGGGG !!" . !! ' ͔- GGGGGG GGGGG SSS . !! ' ͔! GGGGG GGGGGG !!! . !! ' ͔+- GGGGGGGGGG G !!! . !! ' ͔+! GGGGGGG GGGG !!! . !& ' ͔ʗ-$ GGGGGGG GGG GG !!& . !& ' ͔!ʗ GG GGGGGGGGG !!& . !& ' ͔ʗ+-$ GGGGGGGGGGGG !!& . !& ' ͔ʗ+!$ GGGGGGGGGGGG SST . !+ - ' ͔- "" GGGGGGG GGGGGGGGGGGGGGGGGGG !!+ 610 . !+ - ' ͔! "" G GGGGGGGGGGGGGGGGGGGGGGGGG !!+ . !+ - ' ͔+- "" GGGGGGGG GGGGGGGGGGGGGGGGGG !!+ . !+ - ' ͔+! "" GGGGGGG GGGGGGGGGGGGGGGGGGG !!+ . !) - ͔ʗ-$ "" GGGGG GGGGGGGGGGGGGGGGGGGGGGG !!) . !) - ͔!ʗ $ "" GGGGGGG GGGGGGGGGGGGGGGGGGGGG !!) . !) - ͔ʗ+-$ "" GGGGGGG GGGGGGGGGGGGGGGGGGGGG !!) . !) - ͔ʗ+!$ "" GG GGGGGGGGGGGGGGGGGGGGGGGG !!) ! 6 , , 7 GG GGGGGGGGGGG !! ! % , 8 GGGGGGGGGGGGGGGG G !+! & % , 9 GGGGGGGGGGGGGGG G !+& + , ' :8#; GG SUU 60 ) , , ' GGGGGGG GGGGGGGGGGGGGGGGGGGGG G !+) ' (* GGGG !+ . 9 , , ' :8#; GGG G !+ , % , , ' GG GGGGGGGGGGGGGGGGGGGGGGGGGGG !+. 8 GGGG GGGGGGGG !+ % , ͥʚͨʛ GGGGGGGGG GG !+ !" % ͔͚ʚͨʛ GGGGGGG GG !+ !! 9 , ͔͚ʚͨʛTGGGGG G GGG !+ !& % , ͥʗ ʚͨʛTGGGGG GGGG !)" !+ 9 , ͥʗ ʚͨʛTGGGGGGGG GG !)" !) < , GGGGGGGGGGGGG GGGGGGG !)! ! 4 ,͔! GG GGGGGGG !)+ !. % ͔! GGGGG GGGG !)+ ! 4 ,͔! GGGGGG GGG !)) GGGGGGGGGGGGGGGGG 2GGGGGGGGGGGGGGGGGGG )GGGGGGGGG )GGGGGGGGG GGGGGGGGGGGGGGGGGG GGGGGGGGGGGGGGGGGGGGGGGGGGG GGGGGGGGGGG 1 Chapter 1 Literature Review 1.1 Introduction Since the first two-wheeled motorized vehicle was created in 1867 [1], the motorcycle has evolved to become a quite complex machine not only devoted to the task of transportation, but also to leisure and sports. Many kinds, makers, and models are available in the world market nowadays. As an example of this diversity in motorcycle applications, it is noted that cruiser motorcycles, see for instance Figure 1.1, are designed to provide comfort to the rider and stability while driving on a straight path, and their design is intended to allow the rider to travel long distances by limiting fatigue and carrying heavy loads while keeping a stable straight motion. On the contrary, racing motorcycles, see for instance Figure 1.2, are designed to sustain high speeds and to allow the rider to make quick changes in direction in spite of sacrificing stability [43]. Further, off-road motorcycles, see for instance Figure 1.3, require to be light weight and easy to handle, and to have a stronger chassis as well [5]. In all these cases adherence to the road (wheel grip) is an important safety factor, which varies from type to type. Thus, wheel grip and comfort are two features a motorcycle must provide to more or less a degree depending on its potential use. These two requirements are achieved by means of a suspension system, which in general, must provide the following features [26, 93]: I. To isolate the main body of the motorcycle along with the driver from excessive and dangerous destabilizing vibrations produced by the interaction of the wheels and the road roughness; 2 Figure 1.1. Harley Davidson XL1200R Figure 1.2. Suzuki GSXR1000 2010 Roadster (From [207]). (From [208]).