Vehicle Dynamics under Longitudinal Forces - PowerPoint Presentation

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Vehicle Dynamics under Longitudinal ForcesME5670

Date: 22/01/2015

Class timingMonday: 2:30 PM – 4:00 PMThursday: 4:00PM – 5:30 PM

Lecture 5

http://scholar.lib.vt.edu/theses/available/etd-5440202339731121/unrestricted/CHAP3_DOC.pdf

http://scholar.lib.vt.edu/theses/available/etd-5440202339731121

/

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Vehicle

Force and Moment Balance Equations

Wheel

There are two state variables:

Translation velocity: V

Wheel rotational velocity:

Two dynamic equations are

a) Vehicle motion:

b) Wheel motion:Slide3

Tire Tractive (Braking) Force

Tractive (braking ) force is given by

Some interesting phenomena related with tire-ground dynamics under acceleration and deceleration effects:

1. Applying

a driving torque or a braking torque to a pneumatic tire produces

tractive

(braking) force at the tire-ground contact patch

.

2.

The driving

torque produces compression at the tire tread in front of and within

the contact patch. Consequently, the tire travels a shorter distance than it would if it were free rolling.3. When a braking torque is applied, it produces tension at the tire tread within the contact patch and at the front. Because of this tension, the tire travels a larger distance than it would if it were free rolling. This phenomenon is referred as the wheel slip or deformation slip.4. The adhesion coefficient,

, which is the ratio between the tractive (braking) force and the normal load, depends on the road-tire conditions and the value of

the wheel slip, .

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Concept of Wheel Slip

A typical

vs graph is given by  

vs

curve for different road conditions

Wheel slip is defined as

orSlide5

State Equations of Combined System

1. Define convenient state variables:

,

and

2. State equations:

w

here,

,

,

,

,

3. For deceleration:

Wheel slip evolution:

This equation is nonlinear and involves uncertainties in its parameters due to the following reason:

a) Nonlinear relationship of wheel slip and velocity

b) Nonlinear

vs

relationship and depends on road surface .

c) Nonlinear

and

are nonlinear

d)

(Normal tire force) changes based on steering and suspension

,

,

e) Moment of inertia and wind drag force change with time and wind conditionSlide6

Vehicle-Wheel-Surface InteractionSlide7

Local Stability Concept

Local stability: The local stability of a nonlinear system can be studied by linearizing the system

around its equilibrium point. 2. For a given differential equation of the formThe equilibrium equation is stable if the eigenvalues of

have strictly negative real parts.

3. Equilibrium point is given by

4

. Linearized equation is given by

where,

4

. For the unforced state,

i.e.

,

u(t)=0 , the equation is given by

It is unstable

if the eigenvalue of

has positive real part.

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Stability of Vehicle Dynamic Equation

1. Two state equations are

2. Jacobian matrix at the equilibrium

is given by

3. Real part of the eigenvalues of the A matrix are

a) When is positive, the

eigen

values of

A

have negative real parts.

All the values of

b

) When is negative, the

eigen

values of

A

have positive real parts for

u

nder which the system is unstable.Slide9

Practice Problem

Problems based on the chapter 2 and 3 of the reference book titled “Motor Vehicle Dynamics by Giancarlo Genta” (You may use the below mentioned link for Chapter 2, however, full version of chapter 3 can be found from the library)

http://books.google.co.in/books?id=yAo-6yQbg9IC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false Based on the tire-surface interaction, classify different types of forces and moments acting on the tire.Draw the distribution of normal and tangential forces over the contact regions of tire-surface. Discuss the variation of normal forces vs vertical deflection of tire for

radial and bias-ply tire. Finally, also draw the variation of longitudinal force vs deflection, and aligning torque vs rotation.

3. Define the loaded, unloaded and effecting wheel radius with reference to the translational and angular velocity of rolling wheel.4. Define the concept of rolling resistance and write its form in terms of rolling resistance coefficients and normal force.

5. After mentioning the effect of at least four different factors on rolling coefficients, describe

the influence of speed on the rolling coefficients and mention its linear as well as nonlinear

form. Slide10

Practice Problem

6. Define the critical speed of tire and mention its influencing factors.7. Discuss the tractive and braking forces under rolling condition. Also draw the appropriate

curve to describe its variation with slip ratio. 8. Using the magic formula, write the expression of longitudinal forces in terms of slip ratio and also draw its variation for a vertical load of 2, 5, 7, 8 kN.a) Define the cornering force and clearly show the rolling velocity along with the side slip effects. b) After drawing the variation of side force and aligning torque with the slip angle for radial and bias-ply tires, mention the expression of side force and aligning torque

interms of slip angle. c) Also, write the expression of side force and aligning torque based on magic formula.10. Explain the interaction of longitudinal and lateral forces and write the elliptic approximation

of the relationship between these two forces for constant slip angle.Describe and write the expression of the aerodynamic drag forces and moments due to vehicle motion.