Describes all about motorcycle dynamics...
Vehicle System Dynamics 2002, Vol. 37, No. 6, pp. 423±447
0042-3114/02/3706-423$16.00 # Swets & Zeitlinger
A Motorcycle Multi-Body Model for Real Time Simulations Based on the Natural Coordinates Approach VITTORE COSSALTER1 and ROBERTO LOT2 SUMMARY This paper presents an eleven degrees of freedom, non-linear, multi-body dynamics model of a motorcycle. Front and rear chassis, steering system, suspensions and tires are the main features of the model. An original tire model was developed, which takes into account the geometric shape of tires and the elastic deformation of tire carcasses. This model also describes the dynamic behavior of tires in a way similar to relaxation models. Equations of motion stem from the natural coordinates approach. First, each rigid body is described with a set of fully cartesian coordinates. Then, links between the bodies are obtained by means of algebraic equations. This makes it possible to obtain simple equations of motion, even though the coordinates are redundant. The model was implemented in a Fortran code, named FastBike. In order to test the code, both simulated and real slalom and lane change maneuvers were carried out. A very good agreement between the numerical simulations and experimental test was found. The comparison of FastBike's performance with those of some commercial software shows that ®rst is much faster than others. In particular, real time simulations can be carried out using FastBike and it can be employed on a motorcycle simulator.
1. INTRODUCTION The use of computer simulations in motorcycle engineering makes it possible both to reduce designing time and costs and to avoid the risks and dangers associated with experiments and tests. The multi-body model for computer simulations can be built either by developing a mathematical model of the vehicle or by using commercial software for vehicle system dynamics. Even though the ®rst method is more dif®cult and time consuming than the second, maximum ¯exibility in the description of the features of the model can be obtained only by using a mathematical model. In particular, it makes it possible to properly describe the tire behavior at large camber 1
Department of Mechanical Engineering, University of Padova, Italy. Corresponding author: Roberto Lot, Department of Mechanical Engineering, University of Padova, Via Venezia 1, 35131 Padova, Italy. Tel.: 39 049 8276806; Fax: 39 049 8276785; E-mail:
[email protected]; website: www.dinamoto.mecc.unipd.it
2
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V. COSSALTER AND R. LOT
angles, whereas multi-body codes such as ADAMS, DADS or Visual Nastran lack such a feature. Moreover, mathematical modeling has a high computation ef®ciency, while multi-body software require a lot of time to carry out simulations. For the reasons above, the focus of this study was to develop mathematical models of a tire and motorcycle. The tire model properly describes the shape of the carcass and the position of the contact point. Moreover, it takes into account the sliding of the contact patch and the deformation of the tire carcass. The motorcycle model was developed based on the natural coordinates approach [1], which makes it possible to obtain simple equations of motion and hence high computation ef®ciency. 2. MOTORCYCLE AND RIDER DESCRIPTION The motorcycle is modeled as a system of six bodies: the front and rear wheels, the rear assembly (including frame, engine and fuel tank), the front assembly (including steering column, handle-bar and front fork), the rear swinging arm and the unsprung front mass (including fork and brake pliers). The driver is considered to be rigidly attached to the rear assembly; front and rear assembly are linked by means of the steering mechanism. The front suspension is a telescopic type and the rear suspension is a swinging arm type. This vehicle model has eleven degrees of freedom, which can be associated to the coordinates of the rear assembly center of mass, the yaw angle, the roll angle, the pitch angle, the steering angle, the travel of front and rear suspension and the spin rotation of both wheels (see Fig. 1).
Fig. 1. Eleven degrees of freedom motorcycle model.
A MOTORCYCLE MULTI-BODY MODEL
425
The following forces act on the motorcycle elements: suspensions forces due to springs and shock-absorbers, tire forces and torques, aerodynamic forces, rider steering torque, steer damper torque, rear and front brake torques and ®nally propulsive torque, which is transmitted from the sprocket to the rear wheel by means of the chain. The rider's actions on the motorcycle determine both the direction of the vehicle and the forward speed. In this model, the rider is considered to be a rigid body attached to the rear assembly, so that the rider's movement away from the saddle and the corresponding control action are neglected. In this way the motorcycle's direction is controlled only by the torque exerted on the handlebars (steering torque). The forward speed is controlled by applying the brakes (rear and front brake torques) and by acting on the accelerator lever (propulsive force). 3. TIRE MODEL In motorcycles the roll angle can reach 50±55 , hence it has a signi®cant in¯uence both on tire forces and torques and on the contact patch. In this model, the actual shape of the tire is described in detail and the deformation of the tire carcass is taken into account. The road±tire contact is assumed to be dot-shaped and the position of the contact point depends on the roll angle. Tire forces and torques are applied in the contact point. The tire forces include the vertical load N, the lateral force F and the longitudinal force S; the tire torques include the rolling friction torque My and the yaw torque Mz . The tire reference frame Tw is de®ned by using 4 4 transformation matrix notation [2], as shown in Figure 2: its origin is located in wheel center G, plane Xw Zw is the symmetry plane of the wheel, the Xw axis is horizontal and points forwards, the Yw axis is parallel to the wheel spin axis and points rightwards and the Zw axis completes the reference frame. The frame T 0 has its origin located in contact point C, the road plane X 0 Y 0 is horizontal, the X 0 axis is parallel to Xw, points forwards and has unit vector s, the Y 0 axis points rightward and has unit vector n, the Z 0 axis is vertical and points downwards. As it is well known, horizontal tire forces depend on tread deformation and slide, i.e., they depend on sideslip angle l, longitudinal slip k, camber angle j and vertical load N as follows S Sslip
k; l; j; N F Fslip
k; l; j; N
1
In several tire models [3±5] the sideslip angle and longitudinal slip are de®ned according to wheel kinematics, without taking into account the deformation of the tire carcass. On the contrary, in this model slip quantities are de®ned considering the actual contact point, which moves with respect to the rim because of the deformation
426
V. COSSALTER AND R. LOT
Fig. 2. Tire kinematics and tire forces.
Fig. 3. Tire deformability.
of tire carcass. Deformability of the tire carcass is taken into account as shown in Figure 3. The contact point lies on the vertical plane which passes through the wheel spin axis. The tire de¯ection with respect to the rim consists of radial displacement r , lateral displacement l and rotation x around the wheel spin axis. Moreover, it is assumed that tire deformations do not alter the mass properties of the wheel.
A MOTORCYCLE MULTI-BODY MODEL
427
The position of the contact point is expressed by means of its coordinates yc ; zc with respect to frame Tw as follows C Tw f0; yc ; zc ; 1gT
2
Thus, the instantaneous sideslip angle is de®ned as: l ÿarctan
C_ n VY ÿarctan VX C_ s
3
where VX is the forward speed, VY the lateral speed, s and n the unit vectors of axis X 0 and Y 0 respectively. The instantaneous longitudinal slip is de®ned as: k ÿ1 ÿ
_ VR zc
y_ x ÿ1 ÿ VX C_ s
4
where VR is the rolling speed which depends both on spin velocity y_ and rotational _ deformation rate x. On the other hand, tire forces depend on carcass deformation and camber angle, as shown in experimental tests [6, 7] S Selastic
x; j F Felastic
r ; l ; j
5
N Nelastic
r ; l ; j In absence of tire forces, no tire de¯ection is present and the contact point coincides with the point of tangency between the tire surface and road plane C0. Thus, the position of the contact point only depends on the tire shape and the coordinates of C0 with respect to frame Tw can be de®ned as a function of the roll angle, as follows C0 Tw f0; yt
j; zt
j; 1gT
6
where functions yt
j and zt
j make a parametric representation of the lateral pro®le of the carcass. In order to guarantee the condition of tangency between tire and road plane, functions must satisfy the following relation dzt dyt tan
j ÿ dj dj Lateral and radial deformation can be calculated by subtracting expression (6) from expression (2), obtaining l yc ÿ yt
j r zc ÿ zt
j
7
428
V. COSSALTER AND R. LOT
This model is able to properly describe tire behavior both in steady state and transient conditions. Indeed, by coupling Equation (1), which describe the behavior of the contact patch during sliding, with Equation (5), which describe elasticity properties of the tire carcass Sslip
k; l; j; N ÿ Selastic
x; j 0 Fslip
k; l; j; N ÿ Felastic
r ; l ; j 0
8
one obtains a description of tire behavior which is equivalent to relaxation tire models [8±11]. To proof this, let us de®ne a linear relation between longitudinal force and longitudinal slip S Ks k
9
and a linear relation between longitudinal force and rotational deformation S Kx x
10
where Ks and Kx are respectively the longitudinal slip stiffness and rotational stiffness of tire. By substituting Equation (4) in Equation (9) and by rearranging terms, one obtains: ! zc y_ zc x_ zc x_ KS k0 ÿ KS
11 ÿ KS S KS ÿ1 ÿ VX VX VX where k0 is the steady state value of longitudinal slip, which corresponds to the steady state value of longitudinal force S0 : The time derivation of expression (10) yields: S_ x_ Kx
12
By replacing Equations (12) in Equation (11) and by rearranging the terms, one obtains: KS zc =Kx _ S S S0 VX
13
which is a ®rst order relaxation equation, where relaxation length is s KS zc =Kx . The equivalence between this tire model and the relaxation model can be found for lateral force as well. This approach presents several advantages with respect to relaxation models. First, it explains the physical behavior of the tire in more detail, by highlighting both the deformability of the carcass and the sliding of the tread. Furthermore, with this tire model only static and steady state experimental tests are required in order to characterize tire behavior in both static and dynamic conditions.
A MOTORCYCLE MULTI-BODY MODEL
429
In order to complete the model it is necessary to de®ne tire torques with respect to the contact point. The rolling resistance torque is assumed to be proportional to the wheel load
14 My N d where d is the rolling friction parameter. Yaw torque Mz is generated by lateral force F, tire trail t and twisting torque MTz as follows [12±14]:
15 Mz ÿt
lF MTz
j The ®rst term depends on the sideslip angle and tends to align, the second term depends on the roll angle and tends to self-steer. Finally, it is not necessary to take into account overturning moment Mx , because tire forces are applied in the actual contact point [3, 13, 14]. 4. MULTI-BODY MODEL The mathematical model of the motorcycle was developed based on the natural coordinates approach [1]. Natural coordinates consist of cartesian coordinates of points or direction cosines of vectors belonging to the bodies of the system. With this approach, kinematic relationships and equations of motion are very simple. However, the number of variables required for describing a system is larger than the number of degrees of freedom and so additional constraint equations must be introduced. The equations were derived using Maple1, a software which makes it possible to perform symbolic manipulation ef®ciently and to avoid calculation errors. Moreover, it generates automatically the Fortran code. 4.1. Kinematic Description Equations of motion were derived in the inertial reference frame XYZ: axes X and Y are horizontal and lie on the road level, the Z axis is vertical and points downwards; the unit vectors of inertial frame are, respectively, cx , cy and cz . A body-®xed frame Ti is attached to each rigid body. The elements of the transformation matrix are used as generalized coordinates, i.e., the con®guration of each body is described by means of the coordinates of origin and direction cosines of the body-®xed frame (see Fig. 4). The rear tire reference frame Tw1 has its origin in the center of the wheel G1 fx1 ; y1 ; z1 ; 1gT and is de®ned as shown in Section 3, as well as the reference frame T 01 . Moreover, the rear wheel ®xed-frame T1 is obtained from frame Tw1 by a rotation of spin angle y1 around Yw1 axis. It is useful to de®ne the following unit vectors: s1 fsx1 ; sy1 ; 0; 0gT parallel to both Xw1 and X10 axes,
430
V. COSSALTER AND R. LOT
Fig. 4. Description of multi-body system using basic points and unit vectors.
w1 fwx1 ; wy1 ; wz1 ; 0gT parallel to axis Yw1 , v1 fvx1 ; yy1 ; vz1 ; 0gT parallel to axis Zw1 and n1 fÿsy1 ; sx1 ; 0; 0gT parallel to axis Y 01 . The rear assembly ®xed-frame T2 has its origin in the swinging arm pin joint P2 fx2 ; y2 ; z2 ; 1gT ; plane X2Z2 is parallel to plane X1 Z1 , the X2 axis is perpendicular to the steering axis, points forwards and has unit vector u2 fux2 ; uy2 ; uz2 ; 0gT , the Y2 axis has unit vector w2 w1 and ®nally the Z2 axis is parallel to the steering axis and has unit vector v2 fvx2 ; yy2 ; vz2 ; 0gT . The front assembly ®xed-frame T3 has the origin in the point P3 fx3 ; y3 ; z3 ; 1gT , which is the intersection between the steering axis and its perpendicular plane passing through P2 . The X3 Z3 plane is parallel to the symmetry plane of the front wheel, the X3 axis is perpendicular to the steering axis, points forwards and has unit vector u3 fux3 ; uy3 ; uz3 ; 0gT , the Y3 axis is parallel to the front wheel spin axis and has unit vector w3 fwx4 ; wy4 ; wz4 ; 0gT , ®nally the Z3 axis has a unit vector v3 v2 . The front tire reference frame Tw4 has its origin in the center of the wheel G4 fx4 ; y4 ; z4 ; 1gT and is de®ned as shown in Section 3, as well as the reference frame T 04 . Besides, the front wheel ®xed-frame T4 is obtained from frame Tw4 by a rotation of spin angle y4 around Yw4 axis. The following unit vectors are de®ned: s4 fsx4 ; sy4 ; 0; 0gT parallel to both Xw4 and X40 axis, w4 w3 parallel to Yw4 axis, v4 fvx4 ; yy4 ; vz4 ; 0gT parallel to Zw4 axis and n 4 fÿsy4 ; sx4 ; 0; 0gT parallel to Y 04 axis.
A MOTORCYCLE MULTI-BODY MODEL
431
The swinging arm ®xed-frame T5 has its origin in the rear wheel center G1 , the X5 axis is parallel to vector G1 P2 and has unit vector u5 fux5 ; uy5 ; uz5 ; 0gT , the Y5 axis has unit vector w5 w1 and the Z5 axis has unit vector v5 fvx5 ; yy5 ; vz5 ; 0gT . The front unsprung mass ®xed-frame T6 has the origin on the center of mass G6 T4 fGx6 ; Gy6 ; Gz6 ; 1gT ; X6 ; Y6 and Z6 axes are parallel respectively to X3 ; Y3 and Z3 and their unit vectors are u6 u3 , w6 w4 , v6 v2 . The con®guration of the motorcycle is described by means of a set of n 45 coordinates, including the coordinates of points G1 , P2 , P3 , G4 , direction cosines of unit vectors s1 , v1 , w1 , u2 , v2 , u3 , s4 , v4 , w4 , u5 , v5 and spin rotations of both wheels: q fx1 ; y1 ; z1 ; sx1 ; sy1 ; wx1 ; wy1 ; wz1 ; vx1 ; vy1 ; vz1 ; y1 ; x2 ; y2 ; z2 ; ux2 ; uy2 ; uz2 ; vx2 ; vy2 ; vz2 ; x3 ; y3 ; z3 ; ux3 ; uy3 ; uz3 ; x4 ; y4 ; z4 ; sx4 ; sy4 ; wx4 ; wy4 ; wz4 ; vx4 ; vy4 ; vz4 ; y4 ; ux5 ; uy5 ; uz5 ; vx5 ; vy5 ; vz5 gT
16
The motorcycle has only f 11 degrees of freedom, thus it is necessary to formulate a set of m n ÿ f 34 independent constraint equations:
17 fj 0; j 1 . . . m By imposing the unit length condition to all unit vectors, the following 11 independent constraint equations are obtained: f1 s1 s1 ÿ 1 f4 u2 u2 ÿ 1
f2 w1 w1 ÿ 1 f5 v 2 v 2 ÿ 1
f3 v1 v1 ÿ 1 f6 u3 u3 ÿ 1
f7 s4 s4 ÿ 1 f10 u5 u5 ÿ 1
f8 w4 w4 ÿ 1 f11 v5 v5 ÿ 1
f9 v4 v4 ÿ 1
17:1ÿ11
By imposing the orthogonal conditions to every couple of unit vectors which belong to the same reference frame, 15 more independent constraint equations are obtained: f12 s1 w1 f15 u2 w1
f13 s1 v1 f16 v2 u2
f14 v1 w1 f17 v2 w1
f18 u3 v2
f19 u3 w4
f20 v2 w4
f12 s4 w4 f24 u5 w1
f22 s4 v4 f25 v5 w1
f23 v4 w4 f26 w5 v5
17:12ÿ26
The remaining 8 constraint equations are the following: vector G1 P2 must be perpendicular to the rear wheel spin axis Y1 the magnitude of vector G1 P2 must be equal to the swinging arm length lf vector v5 must be perpendicular to the vector G1P2
f27 G1 P2 w1
(17.27)
f28 G1 P2 G1 P2 ÿ l2f
(17.28)
f29 G1 P2 v5
(17.29)
432
V. COSSALTER AND R. LOT
the magnitude of vector P2P3 must be equal to l23 vector P2P3 must lie on the X2Z2 plane (thus it must be perpendicular to the vectors w1 and v2) the point R3 G4 ÿ l1 u3 must lie on the steering axis Z3 (thus it must be perpendicular to vectors w4 and u3)
f30 P2 P3 P2 P3 ÿ l223
(17.30)
f31 P2 P3 w1 f32 P2 P3 v2
(17.31) (17.32)
f33 P3 R3 w4
(17.33)
f34 P3 R3 u3
(17.34)
It is worth pointing out that the natural coordinates approach made it possible to obtain simple constraint equations, which are quadratic with respect to the coordinates. 4.2. Lagrange's Equations Due to the presence of constraints, the Lagrange's equations become m @fj d @K @K X ÿ lj ÿ Qi 0; i 1: : n dt @ q_ i @qi j1 @qi
18
where K is the kinetic energy, li are the Lagrange multipliers and Qi the generalized forces. By coupling the de®nition of kinetic energy to the transformation matrix notation, the kinetic energy of ith rigid body is Z Z 1 1 P_ 2 dm fx; y; z; 1gT_ Ti T_ i fx; y; z; 1gT dm
19 Ki 2 m 2 m where fx; y; z; 1gT are the coordinates of point P with respect to frame Ti . Assuming that the origin of the reference frame is the center of mass of the body and expanding the previous equation, one obtains: 2 3 _i u_ 2i u_ i w_ i u_ i v_ i u_ i G 6 Z _i7 6 w_ i u_ i 7 w_ 2i w_ i v_ i w_ i G 1 7 fx; y; x; 1gT dm fx; y; x; 1g6 Ti 6 7 2 _ 2 m v_ v_ i Gi 5 4 v_ i u_ i v_ i w_ i i
_ i u_ i G _ i w_ i G _ i v_ i _2 G G i Z Z Z Z 1 _2 1 2 1 1 2 2 2 2 dm u_ i x dm w_ i y dm v_ i z2 dm Gi 2 2 2 2 m m m m Z Z Z u_ i w_ i xy dm u_ i v_ i xz dm w_ i v_ i yz dm Zm Zm Zm _ _ _ u_ i Gi x dm v_ i Gi y dm w_ i Gi z dm m
m
m
A MOTORCYCLE MULTI-BODY MODEL
433
By substituting the integral terms in the previous equation with moments and products of inertia with respect to the center of mass, the kinetic energy of each rigid body can be calculated as a function of the elements of transformation matrix Ti , as follows 1 ÿ 1 _2 1 ÿ 2 _ i w_ 2i v_ 2i Iy;i u_ 2i ÿ w_ 2i v_ 2i Ki m i G i Ix;i ÿu 2 4 4 1 ÿ Iz;i u_ 2i w_ 2i ÿ v_ 2i Cxz;i u_ i v_ i Cxy;i u_ i w_ i Cyz;i w_ i v_ i 4
20
If the body center of mass does not coincide with the origin of the reference frame, it _ i T_ i fGxi ; Gyi ; Gzi ; 1gT in the _ i f_xi ; y_ i ; z_ i ; 1gT with G is necessary to replace G previous equation. Thus, the kinetic energy of the whole system is: h i 1 _ 2 1 Iy1 s_ 2 ÿ w_ 2 v_ 2 y_ 1
s1 v_ 1 ÿ s_ 1 v1 y_ 2 1 Id1 w_ 2 K m1 G 1 1 1 1 1 1 2 2 2 ÿ ÿ 1 _ 2 1 Ix2 ÿu_ 2 w_ 2 v_ 2 1 Iy2 u_ 2 ÿ w_ 2 v_ 2 m2 G 2 2 1 2 2 1 2 2 4 4 ÿ 1 Iz2 u_ 22 w_ 22 ÿ v_ 22 Cxz2 u_ 2 v_ 2 Cxy2 u_ 2 w_ 2 Cyz2 w_ 2 v_ 2 4 ÿ ÿ 1 _ 2 1 Ix3 ÿu_ 2 w_ 2 v_ 2 1 Iy3 u_ 2 ÿ w_ 2 v_ 2 m3 G 3 3 4 3 3 4 3 2 4 4 1 ÿ 2 2 2 Iz3 u_ 3 w_ 4 ÿ v_ 3 Cxz3 u_ 3 v_ 3 Cxy3 u_ 3 w_ 4 Cyz3 w_ 4 v_ 3 4 h i 1 _ 2 1 Iy4 s_ 2 ÿ w_ 2 v_ 2 y_ 4
s4 v_ 4 ÿ s_ 4 v4 y_ 2 1 Id4 w_ 2 m4 G 4 4 4 4 4 4 2 2 2 ÿ ÿ 1 _ 2 1 Ix5 ÿu_ 2 w_ 2 v_ 2 1 Iy5 u_ 2 ÿ w_ 2 v_ 2 m5 G 5 5 1 5 5 1 5 2 4 4 1 ÿ Iz5 u_ 25 w_ 21 ÿ v_ 25 Cxz5 u_ 5 v_ 5 Cxy5 u_ 5 w_ 1 Cyz5 w_ 1 v_ 5 4 ÿ ÿ 1 _ 2 1 Ix6 ÿu_ 2 w_ 2 v_ 2 1 Iy6 u_ 2 ÿ w_ 2 v_ 2 m6 G 6 3 4 3 3 4 3 2 4 4 1 ÿ
21 Iz6 u_ 23 w_ 24 ÿ v_ 23 Cxz6 u_ 3 v_ 3 Cxy6 u_ 3 w_ 4 Cyz6 w_ 4 v_ 3 4 where the terms relative to wheels (i 1 and i 4) are slightly different from the terms relative to other bodies because of the axial symmetric structure of the wheels (Ix;i Iz;i Id;i and Cxz;i Cyz;i Cxy;i 0) and because spin velocity y_ 1 ; y_ 4 has been used. The generalized forces expression can be obtained from the virtual work dW of the forces acting on the vehicle m X Qi dqi
22 dW i1
434
V. COSSALTER AND R. LOT
In order to determine virtual works, it is necessary to calculate the virtual rotation dYi of each rigid body with respect to its reference frame Ti . By extending the concept of angular velocity matrix [2] to virtual rotation matrix dY TT dT and by extracting the components of virtual rotation from dY, the following virtual rotation operator can be de®ned: dY
Ti fvi dwi ; ui dvi ; wi dui ; 0gT
23
Virtual work contains the following terms: dW dWg dWS dWA dWt dWB dWt;F dWt;T dWP
24
The virtual work due to the gravity force: dWg
6 X
mi g dGi
24:1
i1
where g f0; 0; g; 1gT is the gravity acceleration. The virtual work due to front suspension force FSf , which acts between the front assembly and front wheel, and virtual work due to rear suspension force FSr , which acts between the rear assembly and swinging arm: dWs FSf v2
dP3 ÿ dR3 ts FSr cy dY
T2 ÿ dY
T5
24:2
where ts @yr =@zr is the velocity coef®cient between spring de¯ection zr and arm rotation yr . The virtual work due to drag, side and lift aerodynamics forces FA Tw1 fFD ; FS ; FL ; 0gT , which are applied on point CA T2 fXCA ; 0; ZCA ; 1gT : dWA FA dCA
24:3
The virtual work due to rider steering torque t and steer damper torque tD , which are applied between the rear and front assembly: dWt
t tD cz dY
T3 ÿ dY
T2
24:4
The virtual work due to rear brake torque MBr , which acts between the rear wheel and swinging arm, and the virtual work due to front brake torque MBf , which acts between the front wheel and front unsprung mass: dWB MBr cy dY
T1 ÿ dY
T5 MBf cy dY
T4 ÿ dY
T6
24:5
The virtual work due to rear tire force FT1 T01 fS1 ; F1 ; ÿN1 ; 0gT , which is applied on rear contact point C1 Tw1 f0; yc1 ; zc1 ; 1gT , and the virtual work due to front tire force FT4 T04 fS4 ; F4 ; ÿN4 ; 0gT , which is applied on front contact point C4 Tw4 f0; yc4 ; zc4 ; 1gT :
dWt;F FT1 dG1 G1 C1 FT1 T1 dY
T1 FT4 dG4 G4 C4 FT4 T4 dY
T4
24:6
A MOTORCYCLE MULTI-BODY MODEL
435
Fig. 5. Geometry of the chain transmission.
The virtual work due to rear tire torque MT1 T01 f0; My1 ; Mz1 ; 0gT and front tire torque MT4 T04 f0; My4 ; Mz4 ; 0gT : dWt;M MT1 T1 dY
T1 MT4 T4 dY
T4
24:7
The virtual work due to the propulsive torque, which is transmitted from the drive sprocket to the wheel by means of the chain. As shown in Figure 5, the drive sprocket center is R T2 fRX ; 0; RZ ; 1gT , whereas the chain angles are: G1 R s1 rc ÿ rp yc1 arctan ÿ arcsin G1 R v1 jG1 Rj G1 R u2 rc ÿ rp yc2 arctan arcsin G 1 R v2 jG1 Rj The chain tension FC T1 fTc sin
yc1 ; 0; Tc cos
yc1 ; 0gT acts between point P7 T1 frc cos
yc1 ; 0; rc sin
yc1 ; 1gT and point P8 T2 RX rp cos
yc2 ; 0; RZ rp sin
yc2 ; 1gT , thus the virtual work is dWp Fc
dP7 ÿ dP8 ÿ Tc rc dy1
24:8
Explicit Lagrange's equations are not shown because of their large number, while their compact form is the following: _ q_ FT k ÿ Q 0 _ q ; k; t M F
q; q; qM
25
where M is the mass matrix, F is the Jacobian matrix of constraint equations (17), k is the vector of Lagrange multipliers and Q is the vector of generalized forces. Due to the natural coordinates approach, the mass matrix is very sparse and has only 9% nonzero elements; moreover the evaluation of Equation (25) require less than 2,000 multiplications and less than 1,000 additions.
436
V. COSSALTER AND R. LOT
4.3. Tire Equations As seen in Section 3, tire deformation is described by means of three coordinates, hence for both the rear and front tires the following six coordinates should be de®ned: q0 fyc1 ; zc1 ; x1 ; yc4 ; zc4 ; x4 gT
26
The tire behavior must be described by means of as many equations as coordinates. Equation (8) can be re-written as follows p1 Sslip;1
k1 ; l1 ; j1 ; N1 ÿ Selastic;1
x1 ; j1 0 p2 Sslip;4
k4 ; l4 ; j4 ; N4 ÿ Selastic;4
x4 ; j4 0 ÿ p3 Fslip;1
k1 ; l1 ; j1 ; N1 ÿ Felastic;1 r;1 ; l;1 ; j1 0 ÿ p4 Fslip;4
k4 ; l4 ; j4 ; N4 ÿ Felastic;4 r;4 ; l;4 ; j4 0
27:1ÿ4
Equations (3), (4) and (7) make it possible to express slip quantities and tire deformations as a function of generalized coordinates, whereas camber angles can be calculated as follows: j1 arcsin
wz1 j4 arcsin
wz4
28
The remaining equations are obtained by imposing the contact between the tire and road plane Z 0, as follows: p5
C1 z z1 wz1 yc1 vz1 zc1 0 p6
C4 z z4 wz4 yc4 vz4 zc4 0
27:5ÿ6
It is worth pointing out that Equations (27.1±4) are differential equations because slip quantities (3) and (4) contain time derivation of coordinates x and x0 . On the contrary, Equations (27.5±6) are algebraic. 4.4. State Space Formulation Lagrange's Equation (25), constraint Equation (17) and tire Equation (27) form a set of 85 second order differential-algebraic simultaneous equations (DAEs) of index 3 [15], with the following unknowns: 51 generalized coordinates and 34 Lagrange multipliers. In order to obtain a 1 index DAEs problem, algebraic constraint Equation (3) should be replaced by differential equations using the Baumgarte stabilization method [16], as follows: 2&o/_ o2 / /0 / where constant and o are properly chosen.
29
437
A MOTORCYCLE MULTI-BODY MODEL
The DAEs problems of index 1 can be numerically solved using the DASSL solver [17], however the transformation of DAEs into a set of ordinary differential equations (ODEs) makes it possible to increase integration speed. For this purpose, the Lagrange multipliers are replaced with the following differential expression: k c t0 c_
30
where constant t0 is properly chosen. Moreover, tire Equation (27) should be replaced by the following set of ODEs p0 fp1 ; p2 ; p3 ; p4 ; p5 t0 p_ 5 ; p6 t0 p_ 6 g
T
31
In addition, the 2nd order Lagrange's Equation (25) should be reduced to a 1st order ODEs. The system is then described by means of the following 2n m 6 130 state variables y fx; v; c; x0 g
T
32
and the following state space equations
9 8 F > > > > = < v ÿ q_ _ t 0 G
y; y; 0 > > > / > ; : p0
33
Although the number of equations is rather high with respect to the number of degrees of freedom, each equation is simple and the evaluation of expression (33) require less than 3,000 multiplications and less than 2,000 additions. These equations have been implemented in a Fortran code, using the implicit solver DASSL for numerical integration. 5. COMPARISON BETWEEN COMPUTER SIMULATIONS AND EXPERIMENTAL MEASUREMENTS In order to validate the multi-body model, some experimental tests were carried out on an Aprilia RSV 1000 motorcycle; they were then compared to the simulation results. The geometrical and inertial characteristics of the motorcycle and the non-linear elastic and damping characteristics of the suspensions were measured at the Department of Mechanical Engineering (DIM) at the University of Padua [18, 19]. Tire parameters were also measured with department's equipment [20], whereas the driver inertia properties were estimated as shown in reference [21]. The characteristics of the motorcycle are given in Appendix and in Figures 10, 11 and 12. The motorcycle was equipped with a measurement system: roll and yaw rate, steering angle, spin velocity of both wheels and steering torque were measured and
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stored on a data recorder [19]. Data post-processing made it possible to calculate vehicle forward speed and roll angle as well. In order to reproduce the experimental maneuvers by means of numerical simulations, steering torque t was calculated according to measured steering torque tm and measured roll angle jm , as follows: t tm kj
jm ÿ j
34
where j is the simulated roll angle and kj the control gain. The chain propulsive force and the front brake torque were calculated based on measured speed um , as follows: 8 < T c r1 S rc : Tc 0
S mr u_ m ku
um ÿ u MFf 0;
S0
acceleration
MFf ÿr4 S;
S
