Basic ground vehicle dynamics 1 Prof. R.G. Longoria Spring 2015
Overview We will be studying wheeled vehicle systems in this course and in the lab, so we ll let that work drive our discussion. Before thinking about how a vehicle becomes automated or cyber-fied, we re going to review key concepts on the ground vehicle mechanics. The approach we will take is to use fundamental laws and principles to build mathematical (systems) models to understand key modes of motion. We want to gain insight into what properties of vehicles to pay attention to when they are in motion motion when driven or piloted by a human and/or by a machine/computing-platform. We will derive vehicle models and to study the dynamic response under prescribed conditions and inputs using computer simulation.
It will be assumed that you have completed an undergraduate course in dynamics, or have sufficient physics background to understand these discussions. This overview will touch on some of those topics and build up to specific needs for vehicle dynamics modeling. Meriam & Kraige (6/3) A two-ale vehicle in acceleration z FBD: FBD: A two-ale vehicle on an incline y z vehicle-fied aes ϕ FBD: with ϕ=0
Concepts from dynamics that underlie vehicle dynamics Coordinate systems used for vehicle motion modeling How to epress position vectors in defined coordinate systems, and how to differentiate to derive velocity and acceleration epressions Particle and rigid body kinematics Relative velocity/acceleration, transferring between coordinate systems Free body (or force) diagrams; static analysis Mass properties (e.g., moments of inertia, inertia matri, etc.) Newton s laws, Euler s equations Models for induced forces and torques, due to friction or elastic effects, as well as applied forces and torques due to eternal sources Coordinate transformations are essential for some problems (e.g., turning) to be reviewed later as need arises Bond graphs (a topic covered in ME 344 and ME 383Q-modeling) can be helpful mostly in powertrain problems, but not necessary for this course
Vehicle-fied coordinate system; SAE standard Ground vehicle coordinate systems commonly employ a coordinate system standardized by SAE*. Consider the standard SAE coordinate system and terminology. *SAE = Society of Automotive Engineers SAE vehicle ais system = forward, on the longitudinal plane of symmetry y = lateral out the right side of the vehicle z = downward with respect to the vehicle p = roll velocity about the ais q = pitch velocity about the y ais r = yaw velocity about the z ais
Earth-fied (or global) coordinate system X = forward travel Y = travel to the right Z = vertical travel (+down) ψ = heading angle (between and X in ground plane) ν = course angle (between vehicle velocity vector and X ais) β = sideslip angle (between and vehicle velocity vector)
Y Eample: a purely kinematic 2D vehicle in inertial frame NOTE: this model will be derived later in class and HW X Let s define the vehicle s kinematic state in the inertial frame by, qi = Y ψ y Velocities in the local (body-fied) ψ reference frame are transformed l 1 l 2 into the inertial frame by the X rotation matri, B or, specifically, qɺ = R( ψ ) qɺ I Inverting, we arrive at the velocities in the global reference X frame, Xɺ U cosψ sinψ 0 v qɺ I = Yɺ = V = ( ψ ) = sinψ cosψ 0 v Ψ qɺ y ψ ɺ Ω 0 0 1 ω z cosψ sinψ 0 R( ψ ) = sinψ cosψ 0 0 0 1 So, for our simple (single-ale) vehicle, the velocities in the inertial frame in terms of the wheel velocities are, qɺ I Rw l2rw ( ω1 + ω2) cos ψ ( ω1 ω2)sinψ 2 B Xɺ Rw l2rw = Yɺ ( ω1 ω2)sin ψ ( ω1 ω2)cosψ = + + 2 B ψ ɺ Rw ( ω1 ω2) B
Eample: Karnopp and Margolis P1.11 NOTE: type of kinematics knowledge that can be helpful in vehicle applications (cf. Karnopp & Margolis, eqs. 1.18) Vp = Vo + Ω R ɺ A = A + Ω R + Ω Ω R p 0 ( ) y w 2 l 2 ω z v y l 1 v δ
Newton s 2 nd Law: You are familiar with this fundamental principle of dynamics, typically stated as f-equals-m-a. It is understood that this equation can be used in different ways. Let s discuss some eamples. My suggestion is that you always think of this law as one that epresses the change in momentum of a body, p, due to imposed forces. Recall definition of (translational) momentum: So Newton s law is: dp dt cause this p = mv dv = m = ma = F dt net forces on a body This defines a dynamic equation. A model. The hard part is figuring out the F.
Sliding and rolling friction need to be understood Friction in sliding and/or rolling plays a critical role in ground vehicle applications. Ogata handout a good review before we begin looking at the tire-terrain interface, which can be very comple. Simple translation: two surfaces sliding, concept of static and sliding friction Friction force as a function of relative velocity Rolling friction vs. rolling resistance. Resistance to motion that takes place when an object rolls over an abutting surface Slip at contact region? Other losses (e.g. in material)? This is a good illustration of the mechanics that lead to rolling resistance.
Eample: 2 ale vehicle, 1 DOF NOTE: you should become comfortable with these types of problems z A two-ale vehicle with mass, m v, at center G. is in maimum acceleration state. Find epressions for the total normal forces at the front and rear pairs of wheels, N f and N r, respectively. Assume the mass of the wheels is small compared with the total mass of the car, and that the coefficient of static friction between the road and the rear driving wheels is µ. FBD: a r f = -component of acceleration m = mass W W = weight on rear wheels = weight on front wheels µ = coefficient of friction h = height of c.g.
Eample: finding reaction forces for two-ale vehicle in maimum acceleration A two-ale vehicle with mass, m v, at center G. is in maimum acceleration state. Find epressions for the total normal forces at the front and rear pairs of wheels, N f and N r, respectively. Assume the mass of the wheels is small compared with the total mass of the car, and that the coefficient of static friction between the road and the rear driving wheels is µ. C Note: since you are not given any specifics about the acceleration, it helps to formulate equations where you can minimize the unknowns. In this case, take moments about C so you don t worry about a. Numerical calculations:
What s going on in the previous slide in partial application of the full 3D equations pruning off what you don t need. The full equations are: F dp = + Ω p dt yz F = pɺ + Ω p Ω p y z z y F = pɺ + Ω p Ω p y y z z F = pɺ + Ω p Ω p z z y y T dh = + Ω h dt yz T = hɺ + Ω h Ω h y z z y T = hɺ + Ω h Ω h y y z z T = hɺ + Ω h Ω h z z y y These are sometimes referred to as the Euler equations, often only when you let the reference aes coincide with the principal aes of inertia at the mass center or at a point fied to the body so the products of inertia go to zero this leads to a simpler form.
Where did they come from? By using body fied coordinates, the rotational inertial properties remain fied. The products of inertia* are all zero, and this makes it convenient for our purposes. *See dynamics tet to review inertial properties. v = v vy vz Ω = Ω Ω Ω y z Define: Then: p mv dp dt XYZ (translational momentum) dp = + Ω p dt With v relative to rotating frame. Newton s law: dp dt XYZ yz = F
State space formulation for dynamic states Fundamentally, these are in momentum states Usually switch to velocity states pɺ = F Ω p + Ω p y z z y pɺ = F Ω p + Ω p y y z z pɺ = F Ω p + Ω p z z y y hɺ = T Ω h + Ω h y z z y hɺ = T Ω h + Ω h y y z z hɺ = T Ω h + Ω h z z y y
By having the full dynamic equations at your disposal, you can: Eamine effects that might be hard to see intuitively or reliably You can throw out terms that do not apply and keep those that will impact the problem at hand. pɺ = F Ω p + Ω p y z z y pɺ = F Ω p + Ω p y y z z pɺ = F Ω p + Ω p z z y y hɺ = T Ω h + Ω h y z z y hɺ = T Ω h + Ω h y y z z hɺ = T Ω h + Ω h z z y y
Static and equilibrium problems Equilibrium problems refer to those that can be solved by assuming there are no dynamic effects. In the case of vehicle dynamics, we can represent this mathematically by making all the rates of change of the 6 DOF zero; i.e., pɺ pɺ pɺ y z = = = 0 0 0 and This would allow us to write up to si equations to solve for any unknown variables. Usually, we need only consider three mutually perpendicular aes, like the body fied aes, but in some cases it may be more convenient to work with other coordinate aes. Also, if the problem is simple enough, the approach does not have to be so general. Effectively, this is what you do when you solve these static problems. hɺ hɺ hɺ y z = = = 0 0 0
Eample: reducing the 6 DOF equations for a 1 DOF vehicle Reduce the Euler equations to the form needed in this problem. Note how you can define the static problems to find unknowns. pɺ = F Ω p + Ω p y z z y pɺ = 0 = F Ω p + Ω p y y z z pɺ = 0 = F Ω p + Ω p z z y y No roll or yaw hɺ = 0 = T Ω h + Ω h y z z y hɺ = 0 = T Ω h + Ω h y y z z hɺ = 0 = T Ω h + Ω h z z y y + F = 0 = W W + W 1 1 z f r W W = f l1 l2 µ h W r 0 About CG: + T = 0 = W l + W l + µ W h y r 2 f 1 r l 2 µ 1 W h f = W and W l r = W L µ h L µ h These two equations give two equations, two unknowns. Solve for the unknown forces, then apply to -direction translational equation to find rate of change of forward velocity (acceleration).
Eample: solving for acceleration of powered mower; given friction Solution from Meriam and Kraige NOTE: at the end, you should feel comfortable with these types of problems However, you should be able to work the problem in a more mature manner, and we will review such approaches later.
Y Apply to 2D vehicle dynamics in an inertial frame Y B l 2 B = track width F Yɺ X wheel force detail (eaggerated angle) = V l 1 m, I z ψ Xɺ NOTE: you can find F = U X X-Y is an inertial frame Lateral forces on wheels constrain side-slip Assume wheels roll without friction Assume small ψ so that, Negligible force at front caster/pivot z z z z 2 ccw U = cosψ v + sinψ v v pɺ = muɺ = mxɺɺ = F = F sinψ Fψ pɺ = mvɺ = myɺɺ = F = + F cosψ + F y y hɺ = I Ω ɺ = I ɺɺ ψ = + T = F l NOTE: we will dig into this model later y The force, F, must satisfy the constraint that there be no lateral slip. This requires, vp = vc + ω rcp V = Y ɺ and the constraint that velocity be zero along rear ale: l 1 V l2ωz Uψ = 0 l 2 C ψ ɺ = Ω P Rear ale U ψ = X ɺ
Summary This is a first look into concepts related to mobility that should become familiar as they are applied in the course. We reviewed concepts from kinematics (motion) and dynamics as applied to ground vehicle problems. The dynamics of a vehicle can important as we try to understand how to make it operate not only as intended, but also when and how it may slip, trip, or rip (i.e., fall apart). Before we can understand how a vehicle can work well with a human or computer driving, we should understand why and when something will work without those influences. Net time, we ll take a detailed look at vehicles in performance mode.
Suggested Reading/References Hibbeler, Engineering Mechanics: Dynamics, 9th ed., Prentice-Hall. Introductory tet often used in course like ME 324. Look for any eamples and problems that have wheels. Karnopp, D.C., and D.L. Margolis, Engineering Applications of Dynamics, John Wiley & Sons, New York, 2008. An ecellent intermediate engineering dynamics tet focused on building mathematical models. Good overview of fundamental material and provides a deeper understanding beyond a traditional sophomore level course in ME dynamics (the road to systems). Meriam, J.L., and L.G. Kraige, Engineering Mechanics: Dynamics (4th ed.), Wiley and Sons, Inc., NY, 1997. Another classical introductory dynamincs tet often used in course like ME 324 (like Hibbeler). Wong, J.Y., Theory of Ground Vehicles, John Wiley and Sons, Inc., New York, 1993 (and later editions up to 2010). One of the current standard books for engineers working in all areas of ground vehicle systems. Relevant but assumes reader has background in fundamental dynamics principles.
Eample Problems The following problems illustrate some of the concepts discussed in these slides. 1. Differential steering of a single-ale vehicle in planar, turning motion (will be discussed later) 2. Finding height of center of gravity of a two-ale (rigid) vehicle 3. Weight distribution in a three-wheeled vehicle on level ground 4. Problem 6/3 (Meriam & Kraige); application of D Alembert Approach
Eample 1: Differential steering of a single-ale vehicle in planar, turning motion Y A U For the simple vehicle model shown to the left, there are negligible forces at point A. This could be a pivot, caster, or some other omnidirectional type wheel.. Assume the vehicle has constant forward velocity, U. Maybe we are interested in the yaw stability of the vehicle about its CG. X Assume the wheels roll without slip and cannot slip laterally. Designate the right wheel 1 and the left 2. What are the velocities in a body-fied frame? v1 = Rwω1 Velocities at v2 = Rwω2 each wheel v = v + v = R ( ω + ω ) ( ) 1 1 2 1 2 2 w 1 2 Since the velocity along the rear ale is constrained to be zero, you can show that, v = l ω w Yaw rate is ω = ( ω ω ) z y R B 2 z 1 2
Eample 1 (cont.): Derivation of the CG velocities vi = vc + You can apply ω rci where i is any point of interest. First, relate the velocity of the left wheel to the CG by, ˆ 2 2 2 ( ˆ ˆ) ˆ ˆ B ˆ B v = v i = v ˆ 2 ( 2 ) ˆ C + ω rc = vi + vy j + ωzk l i + j = v ωz i + vy l ωz j 2 2 B We get the two relations: v2 = v ωz & v2 y = vy l2ωz = 0 2 The second is the lateral constraint and tells us, vy = l2ωz Now for the velocity of the right wheel, ˆ 1 1 1 ( ˆ ˆ) ˆ ˆ B ˆ B v = v i = v ˆ 2 ( 2 ) ˆ C + ω rc = vi + vy j + ωzk l i j = v + ωz i + vy l ωz j 2 2 B So, v1 = v + ωz 2 Adding this relation and the one above yields, v1 + v2 = 2v 1 Rw v = v1 + v2 = ω1 + ω2 Finally, the yaw rate can be found, 2 2 ( ) ( ) 2 2 1 1 1 R ω z = 1 = 1 1 2 = 1 2 = ω1 ω2 B B 2 2 B B w ( v v ) v v v ( v v ) ( )
Eample 1 (cont.): special case of differentially-driven single-ale vehicle with CG on ale For a kinematic model of a differentially-driven vehicle, we assume there is no slip, and that the wheels have controllable speeds, ω 1 and ω 2. If the CG is on the rear ale, Y y ψ B = track width l 1 2 = 0 l = L the velocities in the global reference frame are, X qɺ I Rw ( ω1 + ω2) cosψ 2 Xɺ Rw = Yɺ ( ω1 ω2)sinψ = + 2 ψ ɺ Rw ( ω1 ω2) B Note: this defaults to the common mobile robot model seen throughout robotics literature.
Eample 2: Finding height of center of gravity of a two-ale (rigid) vehicle You can find the height of the center of gravity above the ale length as follows. Assume a rigid suspension. First: ( ) R4 c + d = Wc c = AB CD ( ) = a cosθ H r Then, 4 ( H r) sinθ 4 = ( ) R Lcosθ = Wa cosθ W H r sinθ where :sinθ = Wa R L cotθ W h L
Eample 3: Weight distribution in a three-wheeled vehicle on level ground Top view C d d Right view FBD R R e Front of vehicle l2 1 l L RR LR G G l2 1 G l L W F, F FF F F Taking moments about the R-R ais you can show that, C l + ccw = 0 = = L 2 TR FF L Wl2 FF W Then about C-C (looking from the front), + ccw T = 0 = + We + F d F d C LR RR e ( FRR FLR ) = W d And then about F-F, ( ) ( ) + ccw T = 0 = + Wl F + F L F 1 RR LR l L 1 FRR FLR W + = Solving then for the rear ale forces, F F RR LR e l1 W = + d L 2 l e W L d 2 1 =
Eample 4: Problem 6/3 (Meriam & Kraige); application of D Alembert Approach A bicyclist applies the brakes as he descends a 10 incline. What deceleration a would cause the dangerous condition of tipping about the front wheel A? The combined center of mass of the rider and bicycle is at G. Ans. a = 0.510g