Vehicle Dynamics: Introduction to Motion Analysis

Introduction to Vehicle Dynamics

• Vehicle dynamics studies vehicle motion, focusing on how a vehicle's forward movement changes in response to driver commands.

High-Speed and Road Vehicles

• High-speed vehicles are a primary focus in vehicle dynamics.
• These vehicles typically travel at speeds exceeding 100 km/hr.
• General classifications include ground and fluid vehicles.
• The dynamics of road vehicles are analyzed in this course.
• Road vehicles are affected by road surface conditions, environmental conditions, and driver commands.
• Road irregularities influence the motion of road vehicles.

Formal Definition and Areas of Vehicle Dynamics

• Vehicle Dynamics formally involves studying, measuring, analyzing, and predicting vehicle motion.
• Vehicle dynamics is divided into performance, handling, and ride.

Vehicle Dynamics - Performance

• Vehicle performance relates to its ability to accelerate, decelerate, and maintain speed on various road surfaces in response to driver commands.
• Performance includes studying and reducing fuel consumption for a specific driving cycle.

Vehicle Dynamics - Handling

• Vehicle handling is the directional control and stability of a vehicle in response to steering or environmental inputs.

Vehicle Dynamics - Ride

• Vehicle Ride, or comfort, is the vibratory activity excited by road irregularities and other inputs.

Approach and Decoupling Areas In Vehicle dynamics

• Vehicles are complex dynamic systems, making them difficult to model and interpret when subjected to simultaneous inputs.
• It is common to decouple performance, ride, and handling in most studies.
• Studies can focus on ride motion without control inputs but with environmental inputs.
• Handling is easier to study when assuming a flat road with no irregularities.

Vehicle Handling

• It is an area of vehicle dynamics concerning a vehicle's motion under directional and/or environmental commands.
• Key components of a vehicle system are the driver and the vehicle.

Vehicle Dynamics and Tire Forces

• Vehicle handling dynamics is dependent on the forces the tires produce.

Vehicle Safety Features

• Passive safety features react during a collision to protect occupants.
• Side-impact beams, airbags, high-strength glass, and seat belts are examples of such features
• Active safety features actively try to prevent accidents in many ways.
• Steering control (4WS), traction control, driver-assistance systems, and anti-lock braking systems are examples of active safety features.

Pneumatic Tires - Basics

• A tire (or tyre) is a ring around a wheel's rim.
• It transfers vehicle load and provides traction.
• Materials include synthetic and natural rubber, fabric, and wire, along with carbon black and other compounds.

Tire Construction - Components

• Tires have three primary components:
  • Tread
  • Carcass
  • Bead and bead wires.

Tire Dimensions

• Three main dimensions of pneumatic tires that are: Nominal Section Width (NSW), Nominal Section Height (NSH), Nominal Rim Diameter (NRD).
• The NSW is always specified in tire designations, whereas NSH is not.
• Tire aspect ratio is calculated as NSH/NSW.
• NSH can be determined if the aspect ratio is known.

Tire Cords and Their Impact

• Approaching the circumferential direction, directional control and stability increases
• Approaching the circumferential direction, the ride becomes harsher.
• Radial placement softens the ride.
• Radial placement deteriorates directional control and stability.

Cross Ply Tires

• Cross ply tires first appeared in the 1920s to balance extremes.
• Layers are placed so cords make around a 45-degree angle with a tire's circumferential line.
• Bias angles have been reduced to around 40 degrees, with racing tires using angles as low as 25 degrees.

Radial Ply Tires

• Provides soft ride, good directional control and stability through carcass design.

Tire Forces

• Vehicle behavior depends on forces applied via tire-road contact.
• Study of vehicle handling needs understanding force and moment characteristics of rolling tires.
• The complex and nonlinear dynamic behavior of tires hinders the creation of adequate general mathematical models
• Experimental methods are used to study tire behavior.

Cornering Dynamics

• A moving vehicle resists turning due to inertia, wanting to go straight.
• Tires pull the vehicle, in the direction they are steered.
• There will be a difference between the direction of motion and the steering of the wheels.
• Tires must deform for the contact patch to align with the motion direction.

Contact Patch and Cornering Force Definitions

• Contact patch, or tire footprint, is the tire part contacting the road surface.
• The tire's sideslip angle is the angle between motion direction and its wheel's plane
• Cornering, or lateral force, is the resultant of elastic deformation forces in the contact patch, at right angles to the tire's plane.
• Pneumatic trail notes the distance between lateral force and wheel rotation axis projection on the road.
• Self-aligning torque is force moment that the lateral force exerts reducing the slip angle and aligning direction.

Cornering Force Characteristics

• Tire cornering depends on:
  • the magnitude of the cornering force,
  • vertical tire load,
  • and wheel slip angle, among others.

Cornering Stiffness Defined

• At small slip angles (below 4°), cornering force is a linear function of the slip angle.
• A linear tire model describes the relationship between cornering force and slip angle.
• Cornering stiffness: the derivative of cornering force with respect to slip angle at zero slip angle.

Sign Convention for Cornering

• Slip angle is positive for counterclockwise and negative for clockwise tire rotation during right turns.

Notes on Cornering

• A negative slip angle is needed for a positive lateral force.
• Cornering stiffness is a negative quantity that is required for vehicle handling simulations.

Secondary Factors of Tire Cornering - Inflation Pressure

• Secondary parameters also influence the behavior of tires requiring their effects to be considered
• These including tire inflation pressure, camber angles, and tractive or braking forces.
• Within normal inflation ranges, higher pressure improves cornering by enhancing lateral stiffness.

Secondary Factors of Tire Cornering - Camber Angle

• Camber angle is when the central tire plane deviates from the vertical plane.
• Deviation of the central tire plane is known as camber
• The angle between the planes is called the camber angle.
• A positive angle indicates leaning from the vehicle, a negative indicates leaning to the vehicle.

Effects of Camber Angle on Tires

• Positive camber reduces the cornering force unlike that of a vertical tire.
• Negative camber angle, increases cornering force compared to that of a vertical tire.
• Static camber means the wheels get a camber upon assembly.
• Dynamic camber results from roll motion during cornering or from wheel travel.
• Formula 1 cars use a negative camber angle of -4 degrees in the front, and -2 degrees at the rear.

Secondary Factors of Tire Cornering - Longitudinal Forces

• The cornering force is reduced with the application of longitudinal, tractive, or braking forces.

Friction Circle and Ellipse Defined

• Friction circle concept refers to when that the limiting force is the same for cornering and braking.
• Friction ellipse replaces friction circle when emphasizing traction or braking due to varied force limits.

Self-Aligning Torque

• Application of tractive force raises SAT, braking produces reverse effect.
• SAT increases with slip angle which then then decreases with increasing slip angles.
• SAT may be negative around 15°.
• Normal load on the tire can affect SAT, also leading to an increase.

Tire Models - Purpose

• To study complex tire behavior
• For complex dynamic vehicle models and relating resulting characteristics.

Tire Models - Classification Type

• Models roughly classified:
  • Experimental (empirical): created by fit curves to data.
  • Analytical: simplified physics to allow solvable with governing equations.
  • Physical: use construction material properties with advanced techniques for solution.

Tire Models - Notes

• Empirical and analytical models easily simulate behavior for dynamics models.

Tire Kinematics (Velocity)

• Tires has a velocity of center contact patch in the longitudinal direction
• The velocity =
• For braking velocity increases with Sx
• For traction velocity decreases with Sx

Notes - Tire Kinematics and Longitudinal Slip

• Case 1 velocity @ of tire and radius then the slip = 0 freely rolling tire
• Case 2 velocity not 0 but slip = 1 tire sliding rotational
• Case 3 velocity 0 but rotational then slip = -1 tire slipping while rotating

Slip velocity of the tire

Braking

• Vs² = Vsx² + Vsy²

Traction

• Vs² = Vsx² + Vsy²

Tire Models - Friction Coefficient

• The dependence of tire road friction coefficient, µ, to slip velocity, Vs can be approximated
• Expressed by µ = μo(1 - AsVs)

Where

• µo: peak coefficient of friction
• As: friction reduction factor.
• Vs: slip velocity.

Tire Models - Linear Representation

• Linear is representation of cornering for slip below 4° with Fc = Csα
• Cornering stiffness, Cs, the slope of cornering force
• Valid in area between original and up to 4°
• Should take into account if exceeds these limits

Tire Models - Non-Linear

• Dugoff Model - Used in simulation for qualitative tire representation
• Magic Formula - accurate cornering simulation

Dugoff Non Linear Tire Model

• For combined maneuvers directly, longitudinal and lateral

Formula

• α = slip angle [rad]
• S = longitudinal slip
• Vs = slip velocity [m/s]
• u = velocity component
• µo = static (peak) tire/road friction coefficient
• As = friction reduction factor [s/m]
• Cs = cornering stiffness [N/rad]
• Cl = longitudinal stiffness
• Fx = driving braking, Fy = cornering force and Fz normal load

Dugoff Accuracy

• Poor at higher longitudinal slips

Dugoff Update

• To become better accuracy, you obtain through tire friction that is found through this, (µx=KuMax)

Model: Magic Formula Non Linear Tire Model

• Trig model
• Formula
• (Y= Sy+Dsin Ctan(B(X-Sx)/(1E)+Etan(B(X-Sx) where
• Yany tire response
• X lateral slide
• Six constant must obtain expiremental curve

Simple Non Linear Tire Model

Simple Formula is

Fc = Csa + Aa² Where A is a constant

Taking into account the effect of the tire is longitudinal force during concept F𝑐 = 𝐶𝑠 𝛼 - 4 𝜇 𝑊2

Then as following: 2 types: 𝐹𝐶 = − 𝐶𝑠 𝛼 − 𝑠𝑖𝑔𝑛 𝛼 4 𝜇 𝑊22 1 − 𝜇 𝑊2 lal<ap, 𝐹𝐶 = −𝑠𝑖𝑔𝑛 𝛼 2

2 1 − 𝛼 ≥ 𝛼