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Vehicle Axis Systemes
Introduction
At any given instant of time ,a vehicle subjected to a single force acting at some location and in some direction . This so-called external or applied force maintains the velocity or causes an acceleration of the vehicle .This force is made up of tire,aerodynamic,and gravitational force components. These different components are governed by different physical laws and it is not convenient to deal with this single force.Furthermore ,there separate tire ,aerodynamic ,and gravitational force components act at different locations and in different directions relative to the vehicle chassis.
In order to calculate accelerations and velocities in directions of interest(such as fore and aft for performance or left and right for turning behavior),it is necessary to define axis systems to which the accelerations and velocities and the forces/torques causing them can be referred. Since we live in a three-dimensional world,three reference axes at right angles (90°)to each other which meet at a common point(the axis system”origin”)are the basic and sufficient requirements for these axis systems.
The common axis systems used in vehicle dynamics work in the United States have been defined by the Society of Automotive Engineers(SAE) and appear in Ref.1. Their common use facilitates communication and uniformity of the technical literature .In order to properly relate changes in the motions of the vehicle and its components ,as well as describe the vehicle path relative to the earth ,there are two basic SAE vehicle axis systems.Two additional axis systems are used as required for the presentation of tire force and moment data(described in chapter 2)and aero data(described in Chapter 3).There special axis systems can later be transferred to the vehicle axis system.
Two Types of Axis Systems
The two basic axis systems are described in the following sections.
Earth-Fixed Axis System
This system(the capital letter system) is fixed to the ground and the letters X-Y-Z are used to denote the three principal directions;X and Y are horizontal and at right angles to each cther ,Z is vertical downward.This earth-referenced axis system is used in this text only when it is necessary to reference some aspect of the vehicle motion to a fixed point or direction .In problems related to vehicle motion relative to the ground ,it is common to start off with the origin of the vehicle axis system(described below),coinciding with the earth axis system origin.Frequently ,the axes of the two systems will also coincide initially .The calculation or measurements will then indicate the relative motions of the two axis systems. However ,it must be emphasized that the earth axes location and orientation are independent of the vehicle and are arbitrarily determined at the convenience of the user.
vehicle Axis system
The Vehicle Axis System has its origin in aircraft usage.A review of the reasons for its adoption for automotive studies and the assumptions on which it is based are given below.
The first use of this axis system was by L.Segel in his classic paper,“Theoretical Pre—diction and Experimemal Substantiation of the Response of the Automobile to Steering Control"(Ref.144).The principal reason for its automotive as well as aircraft use is that it is fixed in the Vehicle and moves with it.The inertia properties(moments and proudcts of inertia) remain constant relative to this set of axes but would be variable if referenced to a set of axes fixed on the ground,for example.Not only would they be variable but they would lose in physicaI meaning to the engineer as the Vehicle maneuVered.
Some confusion may arise because this axis system is referred to in different ways. Quite commonly it is caIled a“Moving Axis System”because it moves with the vehicle;it is also labeled as ‘Body axes’it is fixed in the vehicle;frequently it is referred to as the “Stability Axes”or “Directional Control Axis System.” The point to remember is that it is fixed in the vehicle and the inertia properties relative to it are taken constant.
The next questions is: Where is the origin of this axis system located and how are the axes oriented in the vehicle ?This is discussed at length in Ref.111,from which the following is abstracted. The automobile is considered as a two-mass system. The unsprung mass is taken as a rigid frame (with steerable wheels) and the sprung mass is considered as a rigid body .These two masses are “hinged”together at the roll axis and one variable ,roll angle ,specifies the relationship between the two masses .Figure 4.1 (from Ref.111)shows these respective masses and how they are brought together in the complete car.Axes Is and Iu come together in the complete car as i;similarly Js and Ju converge to J,and Ks and Ku to K.
Axes I ,J,and K are relabeled as x,y and z (lower case)in the SAE system shown in Figure 4.2 .The system is orthogonal (the axes are at 90° to each other )and is right-handed (i.e.,a positive rotation about x rotates y into z ,a positive rotation about y rotates z into x,etc.).The x-axis is horizontal and positive forward in the direction of motion when the vehicle is traveling in a straight line on a level road (road is assumed flat).The x-axis lies in the longitudinal plane of symmetry(vehicle assumed to have left-right symmetry).The y-axis point to the driver’s right ,is horizontal and 90°to the x-axis .The z-axis is perpendicular to the other two ,is vertical and positive downward.
The origin of the vehicle axis syatem(x,y,z)is the same for the sprung and unsprung masses,as indicated in Figure 4.1 It is located at the intersection of the vehicle roll axis (shown as a line sloping downward toward the front in Figure 4.1)and a line perpendicular to the road through the CG of the total vehicle ,for a zero roll angle of the sprung mass .
It is usually assumed that the sprung mass rolls about the horizontal x-axis,rather than about the actual roll axis .(Equations have been developed for an inclined roll axis –see Ref.111-but are infrequently used.)Also the origin of the system may,for particular purposes,be located elsewhere.
It is generally assumed that the”principal axes of the rolling mass are parallel to the body axes and that the center of gravty of the non-rolling mass is on the x axis”(Ref.144).
Tire deflections are neglected and the plane of the wheel centers(equal-sized wheels front and rear )remains parallel to the ground.
4.2 vehiele Motions
In vehicle dynamic studies of vehicle motion,it is customary for the user to fix certain operating variables.Thus a value can be assiglled to the forward velocity and the tractiVe/brakingg force or longitudinal acceleration/deceleration.The motion of the unsprung mass can then be studied as perturbation from the steady velocity condition;likewise the roll of the sprung mass can be studied in relation to the unsprung mass.These per-turbations can be initiated by a control action or other disturbance such as a willd gust.For Vehicle stabiIity and control investigations the perturbation Velocities of interest are
Forward velocity u
Lateral velocity v
Yawing velocity r
Rolling velocity p
For stability and control investigation ,the pitch and vertical perturbations (q and w)are neglected. The so-called lateral-directional equations are in terms of v,r and p.
These perturbation velocity(linear and angular)components about the axes fixed in the vehicle must be measured relative to some reference.In aircraft practice it is customary to think in terms of another orthogonal axis system which follows me aircraft through its maneuvers and which at any given instance of time is coincident witll the aircraft system but momentarily fixed in space.The perturbation velocities are measured relative to it.one might say that the reference aXes are continually chasing the aircraft but can conveniently be momentarily stationary when the vehicle velocities are to be measured.This is,of course,equivalent to saying that the perturbation velocities are measured relative to
inertial space.
If actual path along the ground is desired,the Vehicle Axis System is referenced to the Earth-Fixed Axis System as me initial condition.The various acceleration components (rates of change of the Velocity component perturbations)are defined below along with the angular relationships associated with path analysis(see Figure 4.3).
1.Longtudinal Acceleration is the component of the Vector acceleration of a point in the Vehicle in the x—direction.
2.Side Acceleration is the component of the Vector acceleration of a point in the Vehicle in the y-direction.
3.Normal Acceeleration is the component Of me Vector acceleration of a point in the vehicle in the z-direction.
4.LateraI Acceleration is th component of the vector acceleration of a point in the Vehicle perpendicular to the vehicle x-axis and parallel to the road plane.
In steady-state condition,lateral acceleration is equal to the product of centripetal times the cosine of the vehicle's sideslip angle.since in most test condition the sideslip angle is small ,for practical purposes the lateral acceleration can be considered equal centripetal acceleration.
5.Centripetal Acceleration is the component of the vector acceleration of a point in the vehicle perpendicular to the path of that parallel to the road plane.
6.Heading Angle (ψ) is the angle between the trace on the X-Y plane of the vehicle x-axis and the X-axis of the earth-fixed axis system(see Figure 4.3)
7.Sideslip Angle (Attitude Angle,β)is the angle between traces on the X-Y plane of the vehicle x-axis and the vehicle velocity vector at some specified point in the vehicle .Sideslip angle is angle is shown in Figure 4.3 as a negative angle.
8.Course Angle (v)is the angle between the trace of the vehicle vector in the X-Y plane and X-axis of the earth-fixed axis system .A positive course angle is shown in Figure 4.3
Course angle is the sum of heading angle and sideslip angle(v=ψ+β)
图片4.3
9.Vehicle Roll Angle is the angle between the vehicle y-axis and ground plane .
10.Vehicle Pitch Angle is the angle between the vehicle x-axis and the ground plane .
The force/moment components are defined below.
FORCES—The external force acting on the automobile can be summed into one force vector having the following components ;
11.Longitudinal Force(Fx)is the component of the force vector in the x-direction .
12. Side Force (Fy)is the component of the force vector in the y-direction.
13.Normal Force (Fz)is the component of the force vector in the z-direction.
MOMENTS—The external moments acting on the automobile can be summed into one moment vector having the following components:
14.Rolling Moment (Mx) is the component of the moment vector tenging to rotate the vehicle about the x-axis , positive clockwise when looking in the positive direction of the x-axis.
15.Pitch Moment (My) is the component of the moment vector tenging to rotate the vehicle about the y-axis , positive clockwise when looking in the positive direction of the y-axis.
16.Yawing Moment(Mz) is the component of the moment vector tenging to rotate the vehicle about the z-axis , positive clockwise when looking in the positive direction of the z-axis.
4.3 Some Thoughts On Sign Conventions
In the SAE Tire Axis System (see Figure 2.33), the silp angle is defined as the angle between the wheel plane and the direction of wheel travel. In the system , if the wheel is moving forward to the left (as in a right-hand turn ),the slip is negative but the lateral force is quadant .This negative relation between lateral force and slip angle can be confusing in the kinematic relationship between slip and steer angles .
Without proposing a change in the well-established SAE system,this note aims at clarifying some of the issues associated with this sign convention.In the SAE system,slip angle are assumed to be the result of a lateral velocity ,v,in the presence of a forward velocity,u. For exanple , the rear tire slip angle is ?????????(有上标)where Vr is the local lateral velocity.If Vr and u are positive ,the “slip”is to the right , αr is positive ,and lateral force is to the left (negative),corresponding to a left-hand turn . But a slip angle can also be created by steering the wheel, or in aeronautical terminology ,turning the wheel to a yaw angle . In fact, in the SAE system ,a positive steer angle ,δ,produces a positive lateral force .The wheel has been yawed to the right.
Of the two ways of creating an out-of-wheel-plane velocity conlponent,me use of yaw
angle has considerable logic for presenting data from tire tests.We have noted in
Chapter 2,that the term“slip angle”is a questionable one since the wheelis not slipping laterally but rather is operating in a yawed—rolling condition.Defiiling“slip angle’’in erms of a latheral/forward velocity ratio further perpetuates the notion that the tire is iterally sliding sideways as a whole.In actual fact,because of the tire rolling motion,the print has areas composed of adhesion in the front and sliding in the rear. On tire testers,lateral slip velocity is never used to create the out-of-plane velocity:rather the wheel plane is steered(or yawed)relative to the belt velocity.20
If the“slip allgle”convention were relaced by yaw angle,the following would occur:
1.A positiVe yaw angle would line up with a positive rotation in the SAE System
i.e,a clockwise rotation looking forward.It would be compatible with the defhition of positiVe steer angle.
2.In a RH turn,the yaw angles at the wheels would normally be positive as well as the lateral tire forces.The cornering curves would plot in the first quadrant.
3.The aligning torque data would plot in the fourth quadrant and the initial slope of the curve would be negatiVe which is proper for a“stability’’situation.That is,an increase in yaw angle would giVe a negatiVe(or restoring)moment(the tire’s self-aligning torque).
So much for the tire itself,consider the motions of the vehicle and its interaction with the tires.In the SAE vehicle-fixed axis system,a lateral velocity to the right is positive—along the positive y aXis.In a RH turn the vehicle experiences a lateral velocitlI,to the left or negatiVe in this axis system at speeds above the“tangent speed" This aligns with what one senses in the vehicle.Furthermore,in a RH turn the yawing velocity of the vehicle is clockwise and positive—again aligning with one’s senses.The lateral velociy and the yawlng Velocity create lateral velocity components at the front and rear wheels.These velocities,together with the forward velocity,create yaw angle changes at the wheels which in turn account for the vehicle lateral and yaw damping.Thus a vehicle’s lateral velocity to the 1eft(in a right_hand turn)gives rise to lateral force changes to the right,i.e.,the Vehicle damping—in-side slip.A positiVe yawing velocity results in a negative yawing moment,i.e.,the vehicle damping-in—yaw.Without these damping effects a control or disturbance input would result in a continuously accelerated motion。which we know is not the case.
One final thought :SAE J1594 ,”Vehicle Aerodynamics Terminology,”use an axis system which is similar to that of the tire axis system ,except that the out-of-plane velocity vector is defined by a yaw angle ,positive for clockwise rotation .This aligns with aircraft wind tunnel practice .The aerodynamic force and moments all follow aircraft practice and the data falls into the proper quadrant .Damping effects appear with proper signs .
For those who find the signs and piots associated with tire data difficult ,slip angle can be replaced by yaw angles of opposite sign .
4.4 Symbol Conventions in this BOok
In general,we have adopted the SAE Vellicle and Tire Axis Systems(of Ref.1)and their associated symbols—thus X,Y,Z for the Earth一FiXed axes and x,y,z for the Vehicle and Tire Axis.The force(F)and momem(M)components of the Vehicle and Tire Axes are defined by subscripts referencing the particular axis.To distinguish between the vehicle and Tire F/M components we have generally used upper case for the Vehicle subscripts and lower case for tire subscrits,thus Fx,Mx….refer to Vehicle components and Fx,Mx,…。refer to tire components.This is an arbitrary convention.In a book with so many authors and sources,it has been impossible to achieve a uniformity of sysmbol throughout and one must depend on context•In the partlcuIar case of lateral acceleration,we use Ay=ay/g,which again is arbitrary. |
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