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Comparison Between Passive and Active Flexible Anti-Roll Bars Modelled by Finite Element Method on Prevention of Vehicle Rollovers

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Comparison Between Passive and Active Flexible Anti-Roll Bars Modelled by Finite Element Method on Prevention of Vehicle Rollovers COMPARISON BETWEEN PASSIVE AND ACTIVE FLEXIBLE ANTI-ROLL BARS MODELLED BY FINITE ELEMENT METHOD ON PREVENTION OF VEHICLE ROLLOVERS Joed Henrique Paes∗, Diego Colon´ ∗ ∗Av. Prof. Luciano Gualberto, travessa 3, n 158, CEP 05508-900 USP / Escola Polit´ecnica - Laborat´orio de Automa¸c~aoe Controle (LAC) S~aoPaulo, S~aoPaulo, Brazil Emails: [email protected], [email protected] Abstract| Anti-roll bars aims to increase the driving comfort by reducing the roll angle during cornering and also assist the self-steering behaviour of the vehicle chassis. This paper presents a comparison between a passive anti-roll bar and an active anti-roll bar. The bars are modelled by FEM (Finite Element Method) and a controller LQG (Linear Quadratic Gaussian) is designed to the active system. In addition, the vehicle lateral dynamic is hybridized to the flexible model in order to obtain an unique mathematical model. It reviews the vehicle roll dynamics and describes the FEM applied to dynamic modelling of flexible structures. The influence of the active anti-roll system is illustrated through numerical simulation, and a comparison among the passive anti-roll system and the system without anti-roll is also presented. To the best of the authors' knowledge, the modelling approach presented in Sections 3.1 and 3.2 have not been reported in the literature. Keywords| Anti-roll system, flexible structures, control, finite element method. Resumo| Barras de anti-rolagem visam aumentar o conforto atrav´esda redu¸c~aodo ^angulo de rolagem em curvas e tamb´em auxiliam na auto-dirigibilidade do chassis do ve´ıculo. Este trabalho apresenta uma compara¸c~ao entre um sistema com barra de anti-rolagem passiva e outro com barra. As barras s~ao modeladas pelo MEF (M´etodo dos Elementos Finitos) e um controlador LQG (Linear Quadr´aticoGaussiano) ´eprojetado para o sistema ativo. Al´em disso, `adin^amica lateral do ve´ıculo, ´eadicionada a parte flex´ıvel para obter um ´unico modelo matem´atico. O trabalho revisa a din^amica de rolagem ve´ıcular e descreve o MEF aplicado `amodelagem din^amica de estruturas flex´ıveis. A influ^encia do sistema ativo ´eilustrada atrav´esde simula¸c~ao, e uma compara¸c~ao entre os sistemas passivo e sem barra de anti-rolagem tamb´em´eapresentada. At´eonde vai o conhecimento dos autores, a abordagem dada `amodelagem nas Se¸c~oes 3.1 e 3.2 n~aofoi relatada na literatura. Palavras-chave| Sistema anti-rolagem, estruturas flex´ıveis, controle, m´etodo dos elementos finitos. Nomenclature ρ mass density per unit length. _ l lateral distance between left and right sus- () time derivative. pensions. ()n n-order spacial derivative. h height of the vehicle's c.g.. ay lateral acceleration of the vehicle. 1 Introduction ms vehicle sprung mass. The composition of the current automotive fleet mu unsprung mass on vehicle's left/right side. consists of 36% of light trucks, minivans and sport Ixx roll moment of inertia about the c.g.. utility vehicles (SUVs). Unfortunately, the rate of fatal rollovers for pickups is twice the rate for pas- g acceleration due to gravity. senger cars, and the rate for SUVs is almost three Flat total lateral tire force. times the rate for passenger cars. While rollover φ roll angle. affects about 31% of passenger cars involved in crashes with occupant fatalities, it accounts for zs vertical deflection of the vehicle's c.g.. 51% of passenger cars occupant fatalities in sin- zul vertical deflection of the left unsprung mass. gle crashes. These statistics are from 2016 and has motivated several studies to improve passen- zur vertical deflection of the right unsprung mass. ger cars' safety by preventing rollover via active zrl left road displacement input. stability control (NHTSA, 2016). zrr right road displacement input. An active anti-roll system allows the control of the roll angle and, thus, improves the vehicle k suspension stiffness. s stability, especially when turning or moving on bs suspension damping coefficient. slopping ground (Jamil et al., 2017). A rollover kt tire stiffness. is a type of vehicle crash in which a vehicle tips over onto its side or roof and have a higher fa- v nodal displacements. tality rate than other types of vehicle collisions EI bending rigidity. (Phanomchoeng and Rajamani, 2012). This present paper is focused on finding hy- troller for a vehicle, as well as an experimen- bridized dynamic models that merge the car's dy- tal study using the electrically actuated roll con- namic roll movement with a flexible anti-roll bar trol system. Firstly, parameter sensitivity anal- that is attached to the suspension system. This ysis is performed based on the 3DOF linear ve- anti-roll bar can be passive or active. This last hicle model. The controller is designed in the model must be well suited to the design of active framework of lateral acceleration control and gain- feedback controllers, which are used to mitigate scheduled control scheme considering the varying the rollover occurrence. Only linear time invariant parameters induced by laden and running vehicle models are considered. The flexible bar's mathe- condition (Kim and Park, 2004). matical model is obtained by FEM. A method for designing a controller which The suitability for controller synthesis is de- uses an active anti-roll bar (AARB) and an elec- termined by a number of factors, including the tronic stability program (ESP) for rollover pre- accuracy of the model for moderate steering in- vention is presented by Yim. An ESP can carry puts (e.g. linear tire behaviour and small roll out active braking to reduce vehicle's speed and angles) and the order and simplicity of the dy- lateral acceleration to prevent rollover. The con- namic model. The finite element approach pro- troller for the AARB was designed based on linear duces models with large number of degrees of free- quadratic static output feedback control method- dom, which causes numerical difficulties in dy- ology, which attenuates the effect of lateral ac- namic analysis and control design. Thus, a model celeration on the roll angle and the roll rate, by reduction is applied aiming to glean a suitable sys- controling the suspension stroke and the tire's de- tem with lower order to design the LQG controller. flection of the vehicle (Yim et al., 2012). The remainder of the paper is organized as follows: the section 2 presents the results of the 3 Mathematical models literature survey. The section 3 presents the mod- els chosen for this study. The control design is Assuming that the axles and tires have a known presented in the section 4. The section 5 exhibits mass, a four degrees-of-freedom (dof) roll dynamic the results taken from three models: 1) the roll model is developed in the following, considering dynamics model without anti-roll system, 2) with vertical translation of the sprung mass, denoted by passive and 3) active anti-roll bar system as well. zs, vertical translation of the left unsprung mass, Finally, the section 6 presents preliminary conclu- denoted by zul, vertical translation of the right sions from this ongoing study. unsprung mass, denoted by zur, as well as the roll motion of the sprung mass, denoted by φ. The 2 Literature survey Fig. 1 illustrates the modelled system. Carlson and Gerdes use Model Predictive Control zs Flat (MPC) theory to develop a framework for auto- CG P = msg motive stability control. The framework is then φ h demonstrated with a roll mode controller, which ks bs ks bs seeks to actively limit the peak roll angle of the z Roll center z vehicle while simultaneously tracking the driver's ur ul yaw rate command (Carlson and Gerdes, 2003). l Cameron and Brennan present results of zrr k k zrl an initial investigation into models and control t t strategies suitable to prevent vehicle rollover due to untripped driving maneuvers. A challenging Figure 1: Four degree-of-freedom roll dynamics task is identifying suitable vehicle's models from model (source: the authors). the literature, comparing these models with exper- imental results, and determining suitable parame- Rajamani (Rajamani, 2012) shows that the ters for the models (Cameron and Brennan, 2005). overall equations of motion for the model is given A pair of MPC capable of modifying the nom- by: inal roll dynamics of a vehicle through control of the planar vehicle dynamics are presented by Beal msz¨s = ks (2zs zur zul) and Gerdes. Each of these controllers is based − − − (1) bs (2z _s z_ur z_ul) ; on a linear model of the vehicle. Results from − − − a nonlinear vehicle model demonstrate that the l differential-drive technique results in significant murz¨ur = ks zs sin φ zur − 2 − lateral-longitudinal tire force coupling and satu- ration, which degrades the validity of the model l + bs z_s φ_ cos φ z_ur kt (zur zrr) ; used for controller design (Beal and Gerdes, 2010). − 2 − − − Kim presents the design of an active roll con- (2) l mulz¨ul = ks zs + sin φ zul and 2 − m y_2(0; t) m y_2(L; t) k l L = ur + ul t (y(0; t) z )2 + bs z_s + φ_ cos φ z_ul kt (zul zrl) ; B rr 2 − − − 2 2 − 2 − 2 2 (3) kt 2 ksy (0; t) ksy (L; t) (y(L; t) zrl) − 2 − − 2 − 2 and k s l sin φ (y(0; t) y(L; t)) Jsφ¨ =msayh cos φ + msgh sin φ − 2 − + k z (y(0; t) + y(L; t)) : k l2 b l2 s s s sin φ s φ_ cos φ (10) − 2 − 2 (4) ksl bsl + (zul zur) + (_zul z_ur) ; The generalized Hamilton's principle is used 2 − 2 − to derive the equations of motion, i.e.: 2 where Js = (Ixx + msh ).
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