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Development of kineto-dynamic
quarter-car model for synthesis of a
double wishbone suspension
K. P. Balike a , S. Rakheja a & I. Stiharu a
新年快乐 日语a CONCAVE Rearch Centre, Concordia University, 1455 de
Maisonneuve West, Montreal, QC, Canada, H3G 1M8
Version of record first published: 10 Jun 2010.
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Vehicle System Dynamics2014高考试卷
V ol.49,Nos.1–2,January–February 2011,
weld
107–128
Development of kineto-dynamic quarter-car model for
synthesis of a double wishbone suspension
K.P.Balike*,S.Rakheja and I.Stiharu
CONCAVE Rearch Centre,Concordia University,1455de Maisonneuve West,Montreal,
QC,Canada H3G 1M8
(Received 1April 2009;final version received 5October 2009;first published 10June 2010)
Linear or nonlinear 2-degrees of freedom (DOF)quarter-car models have been widely ud to study the conflicting dynamic performances of a vehicle suspension such as ride quality,road holding and rattle space requirements.Such models,however,cannot account for contributions due to suspension kinematics.Considering the proven simplicity and effectiveness of a quarter-car model for such anal-ys,this article prents the formulation of a comprehensive kineto-dynamic quarter-car model to study the kinematic and dynamic properties of a linkage suspension,and influences of linkage geom-etry on lected performance measures.An in-plane 2-DOF model was formulated incorporating the kinematics of a double wishbone suspension comprising an upper control arm,a lower control arm and a strut mounted on the lower control arm.The equivalent suspension and damping rates of the suspen-sion model are analytically derived that could be employed in a conventional quarter-car model.The dynamic respons of the propod model were evaluated under harmonic and bump/pothole excita-tions,idealid by positive /negative rounded pul displacement and compared with tho of the linear quarter-car model to illustrate the contributions due to suspension kinematics.The kineto-dynamic model revealed considerable variations in the wheel and damping rates,camber and wheel-track.Owing to the asymmetric kinematic behaviour of the suspension system,the dynamic respons of the kineto-dynamic model were obrved to be considerably asymmetric about the equilibrium.The propod kineto-dynamic model was subquently applied to
study the influences of links geometry in an attempt to ek reduced suspension lateral packaging space without compromising the kinematic and dynamic performances.
Keywords:kineto-dynamic;suspension;quarter-car model;wheel camber;wheel track;suspension lateral packaging 1.Introduction
Dynamic performances of a vehicle are strongly influenced by its suspension design in a highly complex manner.The synthesis of a vehicle suspension thus involves complex compromis among the various conflicting performance measures.One of the major design compromis is between the ride and handling performance measures,while the ride design involves a *Corresponding author.Email:kb_dia.ca
ISSN 0042-3114print /ISSN 1744-5159online ©2011Taylor &Francis DOI:10.1080/
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108K.P .Balike et al.
further compromi in the occupant comfort and the rattle space requirement [1].Linear or nonlinear
quarter-car models have been widely ud to evaluate the ride,rattle space and the dynamic tyre force respons of suspension design concepts,and synthesis of mi-active and active suspension control strategies,assuming negligible contributions due to suspension kinematics [2–8].It has been suggested that the suspension kinematics can lead to nonlinear respons and affect the vertical dynamics significantly [9,10].The conventional quarter-car model employs equivalent stiffness and damping properties of the suspension coupling the chassis and the wheel mass,while the suspension is permitted to undergo pure vertical deflections.In an independent suspension system,the wheel carrier or the spindle is generally connected to the chassis through the suspension linkages,which induce rotational motion of the wheel apart from the vertical motion.The centre of rotational motion of the wheel relies on the suspension geometry and tends to influence the dynamic respons of the vehicle.Furthermore,the suspension strut is generally mounted away from the unsprung mass centre (cg)and thus the point of application of spring and damping forces and the unsprung mass are not colinear.
The identification of a quarter-vehicle model that incorporates the contributions due to linkage kinematics may thus be desirable for dynamic analys of alternate concepts in an effective manner,and could rve as an effective tool to study the influences of the linkage geometry and joint
flexibilty on the dynamic respons.Such a model is referred to as ‘kineto-dynamic’model in this article.A few studies have attempted to incorporate the suspension kinematics nonlinearities into the dynamic model [9–13].Kim and Ro [9]identified equivalent suspension parameters namely,the sprung and unsprung mass,suspension spring rate and damping rates in compression and rebound of a simple quarter-car model using the respons of a three-dimensional model developed in ADAMS software.The study showed that the identified parameters of a double wishbone suspension vary considerably with changes in the control arms lengths or the strut inclination angle.The parameters of a MacPherson suspension,however,were less nsitive to geometric variations.Similarly,the respons of a nonlinear multibody dynamic MacPherson suspension model were obrved to be comparable with tho of a linear quarter-car model with identified parameters [11].The identification of equivalent parameters of a linkage suspension may require measurements of respons of the physical suspension system which would be cumbersome.Considering the strong effects of variations in the suspension geometry and joint coordinates,such variations in a physical suspension,however,would be extremely demanding.
A few studies have propod kineto-dynamic models of the linkage suspensions,although the vast majority have focusd on the MacPherson suspension.Stensson et al.[13]propod a planar nonlin
ear dynamic model of a MacPherson suspension using the kinematic relations derived from the suspension structure.The dynamic analysis was conducted assuming the chassis as fixed,while the tyre dynamics was neglected.Hong et al.[10]developed a 2-degrees of freedom (DOF)quarter-car model,considering sprung mass vertical displacement and the control arm rotation as the generalid coordinates.The model included the kinematics of the control arm and the strut,while the strut was assumed to be mounted on the control arm.Fallah et al.[14]extended the MacPherson suspension model,propod in [13],by locating the strut on the wheel spindle.The dynamics of the system were derived considering the camber rotation and lateral displacement of the wheel.The study also investigated the variations in the wheel track,and camber,caster and kingpin angles during dynamic events.The above-cited studies on MacPherson suspension quarter-car models considered the tyre as a vertical spring,while the contribution due to its lateral compliance was ignored.The lateral tyre compliance could also influence the dynamic respons particularly when the lateral displacement and the camber rotation of the wheel spindle due to suspension kinematics are considered.Hong et al.
元旦小品剧本[10]and Fallah et al.[14]compared the respons of the kineto-dynamic models with tho D o w n l o a d e d b y [T o n g j i U n i v e r s i t y ] a t 04:10 28 F e b r u a r y 2013
护考120道要对多少题Vehicle System Dynamics 109
of a conventional quarter-car model assuming that the strut positioned on the wheel centre provides equivalency between the two models.Kim and Ro [9]concluded that the contribution of the MacPherson suspension kinematics on the equivalent parameter and the dynamic respons are considerably small.The strut location away from the wheel centre,however,yields some effects of the kinematics.The kinematics of a double wishbone suspension may yield considerably stronger effects on the dynamic respons compared with the MacPherson suspension.This is attributable to kinematics associated with the additional control arm,strut location on the lower control arm and additional kinematic constraints.The identification of an equivalent kineto-dynamic model of a double wishbone suspension has been attempted in a single study,although such a suspension has been widely ud.Joo [15]propod a kineto-dynamic quarter-car model of a double wishbone suspension considering the control arm lengths and angles as the geometric parameters for developing an active control strategy,while the tyre lateral compliance was ignored.This study prents a comprehensive quarter-car kineto-dynamic model incorporating dou-ble wishbone linkage kinematics and tyre lateral compliance in order to study the influences of linkage geometry on the suspension kinematic and dynamic performances.The planar 2-DOF,kineto-dynamic quarter-vehicle
model,is further analyd to derive equivalent suspension and damping rates that may be applied for developing an equivalent quarter-car model.The respons of the kineto-dynamic model are compared with tho of a conventional quarter-car model employing equivalent suspension and damping rates corresponding to static equilibrium under harmonic and idealid bump inputs.The propod model is further ud to synthesi the suspension for lateral packaging requirements,which could be a challenging task in future generation vehicles considering the need for relatively larger sub-frame space for positioning of batteries.2.Model development Figure 1a illustrates the propod kineto-dynamic quarter-car model comprising a double wishbone type of suspension system.The propod planar model includes upper (MN)and lower (OP)control arms,and a strut (AB)mounted on the lower control arm.The strut is modelled assuming linear stiffness and damping properties.The control arms are connected to the chassis and the wheel spindle through revolute joints,while the bushings at the chassis
joints are modelled as linear torsional springs.The control arms are considered to be massless and the total unsprung mass is assumed to be lumped at the centre of gravity (cg)of the wheel asmbly.The tyre is modelled as a combination of a vertical spring and a damper,while the lateral compliance is reprented by lateral linear stiffness.The model is formulated considering vertical disjx
palcements of the sprung mass and the wheel as the DOF as in the ca of a conventional quarter-car model,shown in Figure 1b.
2.1.Kinematic analysis
The variations in the lateral displacement of the wheel spindle and the wheel camber-angle are investigated in terms of displacement respons in the generalid coordinates using the kinematic analysis.The kinematic relations of the planar suspension are formulated using the displacement matrix method [14,16,17],assuming the chassis,the control arms and the wheel spindle as rigid bodies connected through frictionless joints.A coordinate system with its origin fixed at the chassis corresponding to its static equilibrium position is considered.The wheel spindle undergoes translational and rotational motions under a vertical displacement at D o w n l o a d e d b y [T o n g j i U n i v e r s i t y ] a t 04:10 28 F e b r u a r y 2013
110K.P .Balike et
al.
Figure 1.(a)Kineto-dynamic quarter-car model of a double wishbone suspension and (b)conventional quarter-car model.the wheel centre.The wheel spindle displacement along the y -axis directly relates to ‘wheel track variation’,while its rotational motion determines the ‘camber angle variation’.In genera
l,the orientation of a planar rigid body can be determined by the positions of any three points in the plane.For the wheel spindle,the two joint centres N and P ,and the wheel centre C are conveniently chon,where (N y 0,N z 0),(P y 0,P z 0)and (C y 0,C z 0)define the initial coordinates of N ,P and C ,respectively.The first subscript of the variable reprents the coordinate (y or z ),while the cond subscript designates the initial position,when prent.A general displacement matrix for a finite displacement of the wheel spindle in the given plane can be formulated as [17]:[D ]wheelspindle =⎡⎣a 11a 12C y −(a 11C y 0+a 12C z 0)a 21a 22C z −(a 21C y 0+a 22C z 0)001
chine中国china自拍⎤⎦,(1)
where a 11=a 22=cos φ,a 12=−a 21=sin φand φis the wheel spindle rotation about the x -axis or the camber angle.In the above expression,C y and C z are the instantaneous coordinates of the wheel centre C .
The instantaneous coordinates (N y ,N z )and (P y ,P z )of N and P ,respectively,following the application of a wheel spindle displacement z u are derived from the displacement matrix as:
⎡⎣N y P y N z
P z 11⎤⎦=[D ]wheelspindle ⎡⎣N y 0P y 0N z 0
P z 011⎤⎦.
(2)
The above formulation exhibits six unknown parameters corresponding to a given wheel centre vertical displacement (z u ),namely,the y and z coordinates of N and P ,the y -coordinate of C and the wheel rotation angle φ.Equation (2)is solved in conjunction with two constraint conditions impod by the suspension mechanism to obtain kinematic respons.For the planar double wishbone suspension,the constraint equations may be formulated considering D o w n l o a d e d b y [T o n g j i U n i v e r s i t y ] a t 04:10 28 F e b r u a r y 2013
