Tires, Vehicle Dynamics, Aerodynamics and NVH

Introduction and Context

A vehicle is not controlled by its engine alone. Its ability to accelerate, brake, turn, remain stable, provide a comfortable ride, and operate quietly depends on the interaction between the tires, suspension, body structure, air flow, and road surface. These interactions are commonly studied through vehicle dynamics and noise, vibration and harshness (NVH) engineering.

The tire is the vehicle’s only intended contact with the road. It transmits longitudinal forces for driving and braking, lateral forces for cornering, and vertical forces associated with vehicle weight and road irregularity. Aerodynamic forces become increasingly important at high speed, while NVH performance strongly influences perceived quality, safety, and occupant comfort.

This chapter develops mathematical and practical models for tire behavior, load transfer, ride, handling, stability, braking, cornering, aerodynamic drag and lift, and NVH. The models are simplified representations of real vehicles; therefore, their assumptions and limitations must be understood before applying them to design or testing.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Explain tire force generation, slip, friction, compliance, and temperature effects.
2. Calculate static and dynamic axle loads and estimate longitudinal and lateral load transfer.
3. Analyze braking, cornering, ride, handling, and basic stability using suitable models.
4. Estimate aerodynamic drag, downforce, and power demand.
5. Identify NVH sources and interpret measurements using frequency-domain concepts.
6. Relate simulations and test data to design decisions and recognize model limitations.

1. Tire Mechanics and Road-Tire Interaction

1.1 Tire functions and construction

A pneumatic tire supports vertical load, generates longitudinal and lateral forces, absorbs small road irregularities, and provides directional control. Its principal components include the tread, belts, carcass, sidewall, bead, and inner liner. Tread compound and pattern affect friction, wear, water evacuation, and temperature. Sidewall stiffness affects steering response and ride compliance.

The normal tire load is usually denoted by F_z. Longitudinal force F_x acts approximately in the direction of travel, while lateral force F_y acts toward the center of a turn. A tire cannot independently produce unlimited values of both forces. A useful idealized constraint is the friction circle:

Here, μ is the available road-tire friction coefficient. The relation assumes a simple isotropic friction limit and neglects load sensitivity, camber, temperature, and transient effects. Real tires often exhibit a friction ellipse rather than a circle because longitudinal and lateral capacities differ.

1.2 Slip ratio and slip angle

During braking or driving, the wheel rotational speed does not exactly match the vehicle’s forward speed. A commonly used braking slip-ratio definition is:

where V_x is wheel-center longitudinal speed in m/s, R_e is effective rolling radius in m, and ω is angular speed in rad/s. Under this convention, positive κ indicates braking slip. Definitions vary in industry, so the sign convention must always be stated.

Slip angle is the angle between the wheel’s pointing direction and its actual velocity direction. If α is small, the lateral force can be approximated by:

where C_α is cornering stiffness in N/rad. This linear relation is valid only for small slip angles before substantial tire saturation. At larger slip angles, lateral force rises nonlinearly, reaches a peak, and may decline slightly.

1.3 Load sensitivity and combined slip

The friction coefficient is not constant over all loads. As vertical load increases, peak force generally increases less than proportionally. Consequently, distributing load equally between tires can provide more total grip than concentrating it on one tire. This is one reason excessive load transfer reduces total axle cornering capacity.

During combined braking and cornering, part of the tire’s force capacity is already being used longitudinally. For example, if μF_z = 4000 N and F_x = 2400 N, the ideal remaining lateral capacity is:

This is an idealized estimate, not a tire certification value.

1.4 Tire temperature, pressure, and wear

Inflation pressure changes the contact patch shape, sidewall stiffness, rolling resistance, and thermal behavior. Underinflation increases sidewall flexing and heat generation; overinflation can reduce ride comfort and alter contact pressure distribution. Temperature changes pressure and compound properties. Tread wear can reduce wet grip and hydroplaning resistance. Tire analysis must therefore use the specified pressure, load, temperature, and surface condition.

2. Vehicle Loads and Load Transfer

2.1 Static axle loads

Consider a vehicle of mass m, wheelbase L, center-of-gravity height h, and distances a from the front axle and b from the rear axle, where L = a + b. The total weight is W = mg. Static equilibrium gives:

These equations assume a level road, no aerodynamic force, and negligible suspension or tire deformation effects. The axle loads sum to W.

2.2 Longitudinal load transfer

Under acceleration, inertia creates a pitching moment. A simplified rigid-body estimate of longitudinal load transfer is:

where a is longitudinal acceleration in m/s². During acceleration, load transfers rearward; during braking, it transfers forward. Thus, under braking with deceleration magnitude a_b,

The equations assume a flat road, no aerodynamic load, and that the effective force-transfer height is h. Suspension geometry can change the distribution between geometric and elastic load transfer.

2.3 Lateral load transfer

For steady cornering, approximate total lateral load transfer is:

where a_y is lateral acceleration and t is track width. More detailed models distribute this transfer between front and rear according to roll stiffness, suspension geometry, tire stiffness, and center-of-gravity location. Increasing roll stiffness at one axle generally increases that axle’s share of lateral load transfer and can alter handling balance.

Figure: Simplified representation of lateral acceleration, center-of-gravity height, and track-related load transfer.

3. Braking, Cornering, and Handling

3.1 Braking performance

If tire-road friction is the limiting factor and aerodynamic effects are neglected, the ideal braking force is approximately:

The corresponding deceleration is a_b = μg. For a vehicle initially traveling at speed V₀, the constant-deceleration stopping distance is:

This excludes driver reaction distance, brake-system delay, road gradient, tire temperature changes, and aerodynamic drag. Including reaction time t_r, the estimated total distance is:

Anti-lock braking systems regulate wheel slip to remain near the tire’s peak braking region. ABS does not eliminate the laws of friction; it helps preserve steerability and adapts to changing surface conditions.

3.2 Steady-state cornering

For a vehicle negotiating a turn of radius R at speed V, required lateral acceleration is:

The total lateral tire force required is approximately F_y = ma_y. If the road is level and aerodynamic effects are neglected, the minimum ideal radius for a given friction coefficient is:

A real vehicle requires a larger radius because of load sensitivity, steering compliance, tire nonlinearity, road camber, and transient effects.

3.3 Bicycle model and understeer

The linear bicycle model represents the two front tires as one equivalent tire and the two rear tires as another. For small angles and steady cornering, a commonly used relation is:

where δ is steer angle in radians, L is wheelbase, and K is the understeer gradient in rad per g. If K > 0, steer angle increases with lateral acceleration and the vehicle understeers. If K = 0, it is neutrally steering in the linear model. If K < 0, it oversteers; this can produce a critical speed beyond which steady-state control becomes difficult.

The model is useful for interpreting handling tests but is not valid near tire saturation, large body roll, large steering angles, or rapid transients.

3.4 Stability and rollover considerations

Yaw stability depends on tire forces, yaw inertia, steering input, speed, and the distribution of cornering stiffness. Electronic stability control compares intended vehicle motion, inferred from steering angle and speed, with measured yaw rate and lateral acceleration. It selectively applies braking torque and may reduce propulsion torque.

Rollover risk is influenced by track width, center-of-gravity height, suspension compliance, road edge effects, and transient maneuvers. A simple static geometric indicator is the static stability factor:

where t is track width and h is center-of-gravity height. A larger SSF generally indicates greater geometric resistance to rollover, but it is not a complete dynamic rollover prediction.

4. Ride and Suspension Dynamics

4.1 Quarter-car model

A quarter-car model separates sprung mass from unsprung mass. The simplest one-degree-of-freedom ride model consists of sprung mass m_s, suspension spring rate k, and damping coefficient c. Its free-vibration equation is:

The undamped natural frequency is:

and the damping ratio is:

Here, ω_n is in rad/s and ζ is dimensionless. The model assumes linear spring and damper behavior and small displacement. Real dampers are velocity-dependent and often asymmetric between compression and rebound.

A low ride frequency can improve isolation from high-frequency road inputs but may permit excessive body motion. Higher damping controls motion but can transmit more road disturbance. Ride design therefore involves a compromise among comfort, handling, suspension travel, tire contact, and durability.

4.2 Road input and tire hop

The unsprung mass includes the wheel, tire, brake, and part of the suspension. Its natural frequency is higher than the body frequency. Poor control of wheel-hop modes can reduce tire contact during rough-road braking or cornering. A two-degree-of-freedom quarter-car model is commonly used in simulation to examine body bounce, wheel hop, suspension deflection, and dynamic tire load.

5. Aerodynamics

5.1 Drag and lift

Aerodynamic drag is:

where ρ is air density in kg/m³, V is vehicle speed in m/s, C_D is drag coefficient, and A is frontal area in m². Drag power is:

Thus, at constant vehicle specification, drag force varies with speed squared, while drag power varies with speed cubed. Rolling resistance and drivetrain losses must also be included when estimating total propulsion power.

Lift or downforce is similarly represented by:

The sign convention must be stated: positive F_L may denote upward lift, while negative C_L indicates downforce. Aerodynamic balance between front and rear axles is important because it changes normal loads and therefore available tire forces.

5.2 Worked aerodynamic example

An automobile has C_D = 0.30, frontal area A = 2.2 m², speed V = 30 m/s, and air density ρ = 1.2 kg/m³. The drag force is:

The corresponding aerodynamic power is:

This is only the air-drag power at the vehicle speed. Wheel power must additionally overcome rolling resistance, grade, acceleration, and drivetrain losses.

5.3 Design and testing

Aerodynamic development uses computational fluid dynamics, wind-tunnel testing, coast-down testing, pressure measurements, flow visualization, and road correlation. CFD can identify flow structures and compare design alternatives, but turbulence models, moving-ground representation, wheel rotation, mesh quality, and underbody details influence accuracy. Coast-down tests estimate total resistance and require correction for wind, temperature, road gradient, and drivetrain drag.

6. Noise, Vibration and Harshness (NVH)

NVH concerns unwanted sound and motion transmitted through airborne and structure-borne paths. Typical sources include tire-road interaction, powertrain imbalance, combustion orders, gears, electric-machine electromagnetic forces, cooling fans, aerodynamic flow, brakes, and suspension bushings.

6.1 Frequency and resonance

Frequency is measured in hertz, with:

A system can exhibit resonance when excitation frequency approaches a natural frequency. Resonance is not automatically a defect; it becomes a problem when response amplitude, sound pressure, or fatigue exceeds acceptable limits.

For a sinusoidal displacement x(t) = X sin(ωt), velocity amplitude is ωX and acceleration amplitude is ω²X. Therefore, a small displacement at high frequency can produce significant acceleration and perceived harshness.

6.2 Measurement and signal interpretation

Common instruments include accelerometers, microphones, force transducers, tachometers, wheel-speed sensors, and microphones arranged for binaural or sound-power measurements. Time histories reveal events such as impacts, while Fast Fourier Transform (FFT) spectra identify dominant frequency components. Order tracking relates frequency to rotating speed; the nth rotational order is:

where N is rotational speed in rpm and n is the order number. At 1800 rpm, the second order is 2(1800)/60 = 60 Hz.

Frequency-weighted sound levels, such as A-weighted sound pressure level, approximate human hearing sensitivity. Engineering decisions should not rely on a single metric: vibration amplitude, sound pressure, transfer paths, operating condition, and subjective evaluations should be considered together.

6.3 NVH control strategies

Control methods include:

• reducing source excitation through balance, precision gears, tire uniformity, and optimized combustion or motor control;
• interrupting transmission paths using isolation mounts, bushings, damping layers, and tuned absorbers;
• reducing receiver response through acoustic insulation, sealing, active noise control, and cabin treatment.

Active noise control uses sensors and actuators to generate a canceling signal, usually at low frequencies where passive treatment is difficult. It requires stable sensing, suitable phase relationships, and robust operation under changing speed and cabin conditions.

7. Integrated Simulation and Test Practice

A credible engineering workflow begins with a clearly defined operating condition and measurable objectives. A tire model may be identified from force-and-moment testing; a multibody model may represent suspension geometry; a vehicle model may calculate speed, yaw rate, load transfer, and tire utilization; and an aero model may provide drag and downforce coefficients. The outputs should be compared with instrumented road or laboratory tests.

Good practice includes:

1. defining coordinate systems and sign conventions;
2. recording tire pressure, temperature, load, road condition, and vehicle mass;
3. calibrating sensors and checking sampling rates;
4. separating model calibration data from validation data;
5. reporting uncertainty and repeatability;
6. checking conservation of force and moment;
7. avoiding extrapolation beyond the tested range.

Model validation is not merely obtaining a visually similar graph. Error measures, phase differences, transient behavior, and engineering significance must be assessed. For example, a model that predicts steady lateral acceleration correctly but misses yaw-rate overshoot may be unsuitable for stability-control development.

8. Illustrative Case: Improving Wet-Braking Stability

Background

An illustrative midsize front-wheel-drive vehicle shows inconsistent stopping performance on wet roads. Test drivers report steering correction during hard braking, although the ABS warning system indicates normal operation.

Problem analysis

Instrumentation records wheel speeds, steering angle, yaw rate, longitudinal acceleration, tire pressure, and brake pressure. The data show that the left-front tire reaches high slip before the right-front tire. A tire inspection finds unequal tread wear and a 15% pressure difference. The wet-road friction coefficient is also lower than the value used in the original simulation.

Longitudinal load transfer is estimated using ΔF_z = ma_bh/L. The front axle is heavily loaded during braking, but unequal tire conditions cause different peak forces. Because the tire-road force envelope is limited, a tire with reduced wet grip reaches saturation earlier. Small differences in braking force then create a yaw moment.

Approach and solution

The engineering team first equalizes tire pressure and replaces the mismatched tires. It then updates the tire model using wet-surface test data rather than assuming a constant friction coefficient. The ABS calibration is examined for wheel-speed filtering delay and pressure modulation. A vehicle simulation is run with split-friction conditions and validated on a wet test track.

Outcome

The revised calibration reduces yaw deviation while preserving stopping distance. The test plan is expanded to include temperature variation, tire wear, different water depths, and asymmetric surfaces. The improvement is not attributed to ABS software alone; tire condition, sensor quality, hydraulic response, and road friction all contribute.

Learning points

The case demonstrates that vehicle behavior is a system property. A mathematically correct model can still produce poor decisions if its tire parameters are unrepresentative. Test data must document physical conditions, and safety conclusions should be supported by repeatable testing rather than a single run.

9. Limitations, Regulation, and Professional Responsibility

Vehicle tests must comply with applicable braking, tire, lighting, noise, electromagnetic compatibility, and occupant-safety regulations. Requirements vary by jurisdiction and vehicle category; engineers must consult current legal standards rather than rely on informal limits.

Safety-critical algorithms require traceability, controlled changes, verification, and validation. Automated data processing and AI-based models can help classify road noise, identify anomalies, or estimate tire state, but training-data bias, sensor failure, distribution shift, and lack of explainability must be addressed. AI should support—not replace—physical validation and engineering accountability.

Ethically, engineers must report adverse test results, uncertainty, and boundary conditions. Suppressing a failed wet-braking test or presenting an unvalidated simulation as proof of safety can place road users at risk.

Best-Practice Checklist

• Use consistent units, axes, signs, and tire-force conventions.
• Measure rather than assume tire pressure, temperature, mass, and alignment.
• Distinguish static, quasi-static, and transient behavior.
• Include load sensitivity and combined slip when analyzing limits.
• Validate aerodynamic simulations with appropriate physical tests.
• Use both time-domain and frequency-domain NVH analysis.
• Treat model predictions as conditional on their assumptions.
• Preserve test records, calibration information, and uncertainty estimates.

Chapter Summary

Tires generate longitudinal, lateral, and vertical forces through slip and deformation, but their capacity is limited by normal load, friction, temperature, pressure, and combined-slip effects. Static axle loads follow from moment equilibrium, while acceleration and cornering produce longitudinal and lateral load transfer. These changes influence braking, cornering, handling balance, and rollover tendency.

The bicycle model provides a useful first analysis of steering and understeer, while quarter-car models describe fundamental ride and wheel-hop behavior. Aerodynamic drag depends on air density, speed squared, drag coefficient, and frontal area; the associated power rises approximately with speed cubed. Downforce can improve tire capacity but must be balanced between axles.

NVH analysis connects excitation sources, transmission paths, resonances, and human perception. Time histories, FFT spectra, order tracking, and controlled testing help identify problems. Reliable engineering requires calibrated instruments, documented conditions, validated models, uncertainty evaluation, regulatory awareness, and ethical reporting.

Key Terms

Tire slip ratio: Relative difference between wheel circumferential speed and vehicle longitudinal speed.

Slip angle: Angle between a tire’s pointing direction and its actual travel direction.

Cornering stiffness: Approximate slope of lateral tire force with respect to slip angle in the linear region.

Friction circle: Idealized limit relating simultaneous longitudinal and lateral tire forces.

Load transfer: Change in normal load among tires caused by acceleration, cornering, or aerodynamic forces.

Understeer: Handling condition in which steering demand increases with lateral acceleration.

Oversteer: Handling condition in which steering demand decreases with lateral acceleration and rear saturation may dominate.

Yaw rate: Vehicle angular velocity about its vertical axis.

Roll stiffness: Resistance of the suspension and tires to body roll.

Ride frequency: Characteristic oscillation frequency of the sprung vehicle body.

Damping ratio: Dimensionless measure of damping relative to critical damping.

Unsprung mass: Wheel, tire, brake, and suspension mass that moves substantially with the wheel.

Aerodynamic drag: Air resistance opposing vehicle motion.

Downforce: Aerodynamic force acting toward the road.

Frontal area: Projected vehicle area used in aerodynamic force calculations.

NVH: Noise, vibration, and harshness engineering.

Order tracking: Analysis relating vibration frequency to rotating speed.

Resonance: Large response associated with excitation near a natural frequency.

Structure-borne noise: Noise transmitted through solid vehicle structures.

Model validation: Comparison of model predictions with independent measured behavior.

Review Questions

Short Answer Questions

1. What physical quantities determine the idealized friction-circle limit?
2. Why does braking transfer load toward the front axle?
3. Define understeer gradient and state what a positive value indicates.
4. What is the purpose of an FFT in NVH analysis?
5. Why does aerodynamic power increase more rapidly with speed than aerodynamic force?

Long Answer Questions

1. Explain how tire pressure, temperature, wear, and normal load influence vehicle performance.
2. Derive the static front and rear axle loads using moment equilibrium about the axles.
3. Discuss the assumptions and limitations of the linear bicycle model.
4. Explain the operation of a quarter-car ride model and the roles of stiffness and damping.
5. Describe a complete procedure for diagnosing a vehicle vibration using time and frequency data.

Analytical / Application Questions

1. A 1500 kg vehicle has a wheelbase of 2.6 m, center-of-gravity height of 0.55 m, and equal static axle loads. Estimate the front-axle load during braking at 0.8g, neglecting aerodynamic forces.
2. A vehicle travels at 25 m/s on a 100 m radius curve. Calculate the required lateral acceleration and discuss whether a friction coefficient of 0.55 is sufficient in the idealized model.
3. A vehicle has C_D = 0.28, A = 2.0 m², ρ = 1.2 kg/m³, and travels at 35 m/s. Calculate drag force and drag power, then identify two reasons why road-test results may differ from the estimate.

Multiple Choice Questions with Answers

Question 1

Which equation represents the idealized friction-circle constraint?

A. F_x + F_y = μF_z

B. F_x² + F_y² ≤ (μF_z)²

C. F_xF_y = μF_z

D. F_x² − F_y² = μF_z

Correct Answer: B. F_x² + F_y² ≤ (μF_z)²

Question 2

A vehicle traveling at 20 m/s has ideal constant deceleration of 8 m/s². Neglecting reaction time, what is the stopping distance?

A. 20 m

B. 25 m

C. 40 m

D. 160 m

Correct Answer: B. 25 m

Question 3

During hard braking on a level road, the simplified load-transfer model predicts that normal load moves primarily toward the:

A. Rear axle

B. Front axle

C. Inside tires only

D. Outside tires only

Correct Answer: B. Front axle

Question 4

In the linear bicycle model, a positive understeer gradient means that required steer angle:

A. Decreases as lateral acceleration increases

B. Is independent of lateral acceleration

C. Increases as lateral acceleration increases

D. Becomes zero at high speed

Correct Answer: C. Increases as lateral acceleration increases

Question 5

If vehicle speed doubles while all aerodynamic parameters remain constant, aerodynamic drag force ideally becomes:

A. Half as large

B. Twice as large

C. Four times as large

D. Eight times as large

Correct Answer: C. Four times as large

Question 6

Which parameter is measured in N/rad for a linear tire model?

A. Cornering stiffness

B. Friction coefficient

C. Damping ratio

D. Drag coefficient

Correct Answer: A. Cornering stiffness

Question 7

At 1800 rpm, what is the frequency of the second rotational order?

A. 15 Hz

B. 30 Hz

C. 60 Hz

D. 120 Hz

Correct Answer: C. 60 Hz

Question 8

Which change generally increases the geometric static stability factor?

A. Increasing center-of-gravity height

B. Reducing track width

C. Increasing track width

D. Reducing wheelbase only

Correct Answer: C. Increasing track width

Question 9

What is the principal purpose of anti-lock braking control?

A. To eliminate aerodynamic drag

B. To maintain useful wheel slip and preserve steerability

C. To increase tire temperature deliberately

D. To make all tires carry equal static load

Correct Answer: B. To maintain useful wheel slip and preserve steerability

Question 10

Which method is most directly associated with identifying dominant vibration frequencies?

A. FFT spectrum analysis

B. Static axle weighing only

C. Paint-thickness measurement

D. Fuel-consumption averaging only

Correct Answer: A. FFT spectrum analysis