Vehicle Mechanics, Dynamics, and Performance

Introduction

Electrified-vehicle design begins with a physically correct description of how a vehicle interacts with the road, air, and environment. The battery, inverter, motor, transmission, brakes, and thermal systems can be sized only after the required tractive force and power are known. This chapter develops the fundamental models used to calculate road load, acceleration, gradeability, braking, drive-cycle energy consumption, range, and major propulsion-component ratings.

The methods apply to battery-electric vehicles (BEVs), hybrid-electric vehicles (HEVs), and plug-in hybrid-electric vehicles (PHEVs). The central principle is conservation of force and energy: propulsion must supply road-load forces and acceleration demand, while braking systems must safely remove kinetic and potential energy.

Learning Objectives

By the end of this chapter, students should be able to:

1. Resolve the principal forces acting on a vehicle.
2. Calculate rolling resistance, aerodynamic drag, grade force, inertial force, tractive force, and wheel power.
3. Determine acceleration, gradeability, top speed, and braking performance.
4. Interpret drive cycles and estimate energy consumption and range.
5. Account for drivetrain efficiency, regenerative braking, auxiliary loads, and battery limits.
6. Select preliminary ratings for motors, batteries, inverters, and braking systems.
7. Recognize assumptions, uncertainties, safety constraints, and limitations of vehicle-performance models.

1. Vehicle as a Longitudinal Dynamic System

For preliminary analysis, a vehicle is commonly represented as a rigid body moving along a road. The longitudinal direction is the direction of travel. The principal inputs are wheel tractive force, road slope, wind, and braking force. Outputs include speed, acceleration, power, energy consumption, and stopping distance.

Important terms include:

• Mass, m: vehicle mass in kilograms, including occupants and payload.
• Speed, v: vehicle speed in metres per second.
• Acceleration, a: rate of change of speed in metres per second squared.
• Tractive force: force delivered at the driven tyre contact patches.
• Road load: the sum of rolling resistance, aerodynamic drag, and grade resistance.
• Drive cycle: a prescribed speed-versus-time profile used for comparison or simulation.
• State of charge (SOC): stored battery energy expressed relative to usable battery capacity.

A free-body representation is useful because it separates forces that are speed-dependent, slope-dependent, and acceleration-dependent.

Figure: Simplified longitudinal free-body diagram for a vehicle climbing a grade.

Taking the positive direction uphill and forward, Newton’s second law gives:

Here, Fₜ is tractive force at the wheels, Fᵣ is rolling resistance, Fₐ is aerodynamic drag, Fg is grade force, mₑ is equivalent inertial mass, and a is acceleration. The equation assumes a one-dimensional model and neglects lateral dynamics, suspension transients, and tyre deformation beyond their representation in the resistance terms.

2. Road-Load Forces

2.1 Rolling Resistance

Rolling resistance arises mainly from tyre deformation and contact-patch hysteresis. A widely used engineering approximation is:

where Cᵣ is the rolling-resistance coefficient, m is vehicle mass, g is gravitational acceleration (approximately 9.81 m/s²), and θ is road angle. On level roads, cos θ ≈ 1, so Fᵣ ≈ Cᵣmg. Values depend on tyre construction, inflation pressure, temperature, load, and road surface. Using a constant coefficient is suitable for preliminary calculations but not for detailed tyre modelling.

2.2 Aerodynamic Drag

Aerodynamic drag is:

where ρ is air density in kg/m³, Cᴅ is the drag coefficient, A is frontal area in m², and vᵣ is relative air velocity in m/s. With wind speed vw defined positive in the travel direction, a consistent convention is vᵣ = v − vw. Drag power increases approximately with the cube of speed because Pₐ = Fₐv and Fₐ ∝ v². This explains why high-speed driving is particularly energy-intensive.

2.3 Grade Force

For a road angle θ, the component of vehicle weight parallel to the road is:

Road grades are often stated as a percentage:

For modest grades, sin θ ≈ tan θ ≈ grade/100. Thus a 10% grade corresponds approximately to a 0.10 fraction of horizontal distance, but the exact angle is θ = arctan(0.10). Grade force is independent of speed, although the power required to climb it increases with speed.

2.4 Inertial Force and Equivalent Mass

Acceleration requires force:

The equivalent mass includes translational vehicle mass and the rotational inertia of wheels, shafts, gears, and motor rotors. A simplified model uses:

where δ is an inertial-mass factor slightly greater than one. Alternatively, each rotating component can be reflected to the vehicle through its gear ratio. For early design, δ = 1.03–1.10 may be used cautiously; the value should be validated against the actual drivetrain.

2.5 Total Road Load

The total force needed at the tyres is:

At constant speed, a = 0. On a descent, θ is negative and the grade term assists forward motion. The model does not automatically include bearing losses, driveline losses, tyre slip, or accessories; these must be represented separately when greater accuracy is required.

3. Wheel Torque, Motor Torque, and Power

If the effective tyre radius is rᵥ, wheel torque is:

For a gear ratio i = ωm/ωw and drivetrain mechanical efficiency ηg, motor torque required during propulsion is approximately:

Motor speed is:

where angular speed is in rad/s. If motor speed is required in revolutions per minute:

Wheel power is:

Motor shaft power during propulsion is approximately:

Electrical battery power must also include motor and inverter efficiency and auxiliaries. If the combined propulsion efficiency from battery terminals to wheels is ηp, then:

where Paux is auxiliary electrical power. During regenerative braking, the battery receives only a fraction of mechanical braking power:

The sign convention must be stated explicitly because software packages differ in whether charging power is positive or negative.

4. Performance Measures

4.1 Acceleration

Given available tractive force Favailable, acceleration is:

where Froad,non-inertial includes rolling, aerodynamic, and grade forces but excludes inertial force. Available force is limited by both the powertrain and tyre-road adhesion:

Here μ is the tyre-road friction coefficient and Ndrive is the normal load on driven tyres. For an idealized level-road vehicle with a fixed load distribution, this is only an approximation. During acceleration, load transfer changes axle normal forces, and suspension geometry affects the result.

4.2 Gradeability

At constant speed up a grade, acceleration is zero. If available wheel force is Fₜ,max, the maximum grade angle satisfies:

This equation should be solved for θ. A common small-angle approximation is:

but aerodynamic drag must not be omitted at high speed.

4.3 Maximum Speed

On level ground at maximum speed, power available equals road-load power:

Because this equation contains both v and v³, maximum speed is generally found numerically or by solving the resulting cubic. Motor maximum speed, battery power, tyre rating, thermal limits, and legal constraints may impose lower limits.

4.4 Braking and Stopping Distance

For constant deceleration aᵦ on a level road, stopping distance from initial speed v₀ is:

Reaction distance for driver or controller delay tr is:

Thus total stopping distance is:

These equations assume constant deceleration and do not include brake-system response variation, grade, tyre slip, or aerodynamic effects. On a downhill road, gravity increases stopping demand; on an uphill road, it assists stopping.

The kinetic energy that must be dissipated or recovered is:

A vehicle travelling at twice the speed has four times the kinetic energy. Regenerative braking cannot recover all of this energy because of motor, inverter, battery, tyre, and control limitations. Mechanical friction brakes must provide the remaining braking torque and must be capable of stopping the vehicle if the battery is full or unavailable.

4.5 Regenerative Braking Constraints

Regeneration is limited by:

• maximum negative motor torque;
• inverter current and voltage limits;
• battery charge-power acceptance;
• battery temperature and SOC;
• tyre-road adhesion;
• axle load transfer;
• minimum speed below which the motor cannot provide effective regeneration;
• brake blending and functional-safety requirements.

A robust brake controller requests the required deceleration, allocates allowable regenerative torque, and commands friction brakes for the residual torque. The transition should be smooth and predictable to preserve pedal feel and vehicle stability.

5. Drive Cycles and Energy Consumption

A drive cycle is a time sequence v(t). The vehicle model calculates acceleration, road-load force, power, and energy at each time step. Acceleration is obtained from:

For discrete data with sample interval ∆t, a simple approximation is:

At each step, wheel power is:

If Pwheel,k > 0, battery discharge energy is approximately:

If Pwheel,k < 0, the magnitude of regenerative energy accepted by the battery is bounded by the regenerative efficiency and battery charge-power limit. The net battery energy is the discharge energy minus accepted recovered energy, subject to the chosen sign convention.

Energy consumption is commonly reported as kWh/100 km:

where Ebatt is battery energy in kWh and distance is in kilometres. Testing conditions must identify vehicle mass, tyre pressure, temperature, wind, auxiliary loads, cycle definition, and whether charging losses are included. “Wall-to-wheel” consumption includes charger losses; “battery-to-wheel” consumption does not.

Worked Example 1: Level-Road Tractive Power

An EV has mass 1,800 kg, Cᵣ = 0.012, Cᴅ = 0.28, frontal area 2.2 m², and travels at 90 km/h on a level road. Assume air density 1.225 kg/m³, no wind, and propulsion efficiency 0.90. Convert speed:

v = 90/3.6 = 25 m/s.

Rolling resistance:

Fᵣ = 0.012 × 1800 × 9.81 = 211.9 N.

Aerodynamic drag:

Fₐ = 0.5 × 1.225 × 0.28 × 2.2 × 25² = 236.1 N.

Total steady-state road load is approximately 448 N. Wheel power is:

Pwheel = 448 × 25 = 11.2 kW.

Battery propulsion power, excluding auxiliaries, is:

Pbatt = 11.2/0.90 = 12.4 kW.

The result demonstrates that even on level ground, aerodynamic drag is comparable with rolling resistance at this speed. At higher speeds, drag becomes dominant.

Worked Example 2: Acceleration on a Level Road

Use the preceding vehicle at 50 km/h, with available wheel force 4,000 N and inertial-mass factor δ = 1.05. Let v = 13.89 m/s. Rolling resistance is 211.9 N. Aerodynamic drag is approximately:

Fₐ = 0.5 × 1.225 × 0.28 × 2.2 × 13.89² ≈ 72.9 N.

Thus:

mₑ = 1.05 × 1800 = 1890 kg

and

a = (4000 − 211.9 − 72.9)/1890 ≈ 1.97 m/s².

The calculation assumes that the available force is below the tyre-adhesion limit and that motor torque is available at the corresponding speed.

6. Range and Component Sizing

6.1 Usable Battery Energy and Range

If usable battery energy is Eusable and average battery consumption is e, estimated range is:

If nominal battery energy is Enom and only a SOC window from SOCmax to SOCmin is used, a first approximation is:

where ηusable represents additional usable-energy and conversion constraints. Real range varies with speed, climate, payload, terrain, battery temperature, ageing, and accessory use. A single range number is therefore incomplete without test conditions.

6.2 Battery Energy and Power Ratings

Battery energy rating should satisfy the required mission energy with reserve:

Battery power rating must satisfy peak discharge and charge demands:

The battery must also satisfy voltage and current constraints:

At a given power, a higher pack voltage reduces current, which generally reduces conductor losses and permits smaller cables, although insulation, switching, creepage, clearance, and safety requirements become more demanding.

6.3 Motor and Inverter Sizing

Continuous motor power is determined by sustained requirements such as highway cruising and long grades. Peak power is determined by acceleration, overtaking, and short-duration grade demands. Motor torque must cover low-speed launch requirements, while motor speed must cover the maximum wheel speed through the gear ratio.

The inverter must be rated for motor phase current, DC-link voltage, switching temperature, short-duration overload, and fault conditions. Ratings should be based on duty cycle and thermal analysis rather than peak numbers alone.

6.4 Hybrid Power Allocation

In an HEV, the engine and electric machine share propulsion demand:

where Pe is engine output power, Pm is motor output power, and Ploss includes transmission and conversion losses. The supervisory controller selects an operating point considering fuel economy, emissions, battery SOC, drivability, thermal limits, and component ageing. A smaller battery may be adequate for an HEV because the engine supplies sustained energy, whereas a BEV battery must support the complete electric mission.

7. Illustrative Integrated Case: Urban BEV Sizing

Background

The following is a fictional illustrative case for a compact BEV intended for urban and suburban operation. The design mass is 1,600 kg including occupants, Cᵣ = 0.011, Cᴅ = 0.30, frontal area 2.1 m², tyre radius 0.30 m, and combined battery-to-wheel efficiency 0.88. Auxiliary demand is 1.0 kW in mild conditions.

Problem

The vehicle must cruise at 100 km/h on level ground, climb a 6% grade at 60 km/h, and achieve a 0–50 km/h launch with an average acceleration of 2.5 m/s². Determine approximate wheel-force and power requirements.

Analysis

At 100 km/h, v = 27.78 m/s. Rolling resistance is:

Fᵣ = 0.011 × 1600 × 9.81 ≈ 172.7 N.

Aerodynamic drag is:

Fₐ = 0.5 × 1.225 × 0.30 × 2.1 × 27.78² ≈ 296.4 N.

Total force is approximately 469 N, so wheel power is:

Pwheel ≈ 469 × 27.78 = 13.0 kW.

Battery power including auxiliaries is approximately:

Pbatt = 13.0/0.88 + 1.0 ≈ 15.8 kW.

For the 6% grade, use sin θ ≈ 0.06 and cos θ ≈ 1. At 60 km/h, v = 16.67 m/s. Rolling resistance is approximately 172.7 N, aerodynamic drag approximately 106.7 N, and grade force approximately:

Fg = 1600 × 9.81 × 0.06 ≈ 941.8 N.

Total wheel force is approximately 1,221 N, giving:

Pwheel ≈ 1221 × 16.67 = 20.4 kW.

Battery power is approximately:

Pbatt ≈ 20.4/0.88 + 1.0 = 24.2 kW.

For the launch, average inertial force is:

Fᵢ = 1600 × 2.5 = 4000 N.

At an average speed of 25 km/h, resistance is modest; assume approximately 210 N. Required wheel force is therefore about 4,210 N. Wheel torque is:

Twheel = 4210 × 0.30 ≈ 1263 Nm.

A single-speed reduction of ratio 9:1 with 95% mechanical efficiency would require motor torque of approximately:

Tm = 1263/(9 × 0.95) ≈ 148 Nm.

This is an average launch requirement. Peak torque should be higher to accommodate initial conditions, road variation, and control margin.

Approach and Outcome

A preliminary design could therefore target at least 25 kW continuous electrical power for the specified grade, a substantially higher short-duration peak power for acceleration, and a motor capable of approximately 150 Nm or more at low speed after applying a suitable engineering margin. Final ratings require simulation across the complete drive cycle, thermal testing, tyre-adhesion analysis, battery voltage limits, and regulatory braking validation.

Learning Points

1. Gradeability can demand more power than level-road cruising even at lower speed.
2. Launch performance is usually torque-limited rather than power-limited at very low speed.
3. Continuous and peak ratings must be separated.
4. Auxiliary loads become significant when propulsion power is small.
5. Preliminary equations support architecture decisions but do not replace validated vehicle simulation.

8. Modelling Practice, Limitations, and Professional Responsibility

A credible model documents every assumption: mass definition, tyre condition, air density, wind convention, road grade, efficiency map, auxiliary load, time step, regenerative-braking strategy, and battery SOC limits. Unit conversion errors are common, especially confusion between km/h and m/s, kW and W, and percentage grade and angle.

Constant efficiency is useful for hand calculations but inaccurate over a broad operating range. Motor and inverter efficiency vary with torque, speed, temperature, and DC voltage. Battery energy depends on current, temperature, SOC, ageing, and allowable voltage limits. Road-load coefficients also vary with tyre pressure and surface.

Vehicle safety has priority over energy efficiency. Brake blending must preserve required deceleration even when regeneration is unavailable. High-voltage systems require isolation monitoring, interlocks, controlled discharge, crash protection, and competent maintenance procedures. Performance claims should be traceable to defined test procedures rather than selectively chosen conditions. Data-driven and AI-based predictors may improve energy estimation, but they require representative data, validation over changing weather and payload, and safeguards against unsafe control decisions.

Contemporary development increasingly combines one-dimensional vehicle models, motor and battery maps, hardware-in-the-loop testing, connected traffic information, and predictive energy management. These tools improve accuracy but also increase the importance of model governance, cybersecurity, calibration control, and explainable safety decisions.

Chapter Summary

Vehicle performance analysis is founded on the longitudinal force balance between tractive force, rolling resistance, aerodynamic drag, grade force, and inertial force. Rolling resistance is approximately proportional to weight, aerodynamic drag varies with the square of relative air speed, and grade force is proportional to the sine of road angle. Wheel power equals tractive force multiplied by speed, while battery power must include conversion losses and auxiliaries.

Acceleration, gradeability, top speed, braking, and range follow from the same physical model but require different boundary conditions. Regenerative braking can reduce energy consumption but is constrained by motor, inverter, battery, tyre, thermal, and control limits. Drive-cycle analysis integrates instantaneous power over time and must clearly state test conditions. Component sizing distinguishes continuous from peak requirements and must include electrical, thermal, mechanical, safety, and ageing margins.

Key Terms

Road load: Total force opposing or requiring vehicle motion.

Rolling resistance: Tyre and contact losses opposing motion.

Aerodynamic drag: Air resistance caused by relative airflow.

Grade force: Weight component parallel to a sloped road.

Tractive force: Propulsive force at the tyre-road interface.

Equivalent mass: Translational mass augmented by reflected rotational inertia.

Drag coefficient: Dimensionless measure of aerodynamic resistance.

Frontal area: Vehicle projected area normal to airflow.

Wheel torque: Torque applied at the driven wheels.

Drive cycle: Prescribed vehicle speed as a function of time.

Wheel power: Mechanical power delivered at the tyres.

Battery power: Electrical power drawn from or delivered to the battery.

Regenerative braking: Conversion of vehicle kinetic energy into stored electrical energy.

Gradeability: Ability to climb a specified slope at a specified speed.

SOC: Battery state of charge.

Usable energy: Energy available within permitted battery operating limits.

Peak power: Short-duration maximum power capability.

Continuous power: Power sustainable under specified thermal conditions.

Brake blending: Coordinated use of regenerative and friction braking.

Wall-to-wheel efficiency: Efficiency including energy supplied from the charging source through to vehicle motion.

Review Questions

Short Answer Questions

1. What are the three principal steady-speed road-load forces?
2. Why does aerodynamic drag become increasingly important at high speed?
3. Define gradeability.
4. What is the purpose of brake blending in an electrified vehicle?
5. Distinguish between battery-to-wheel and wall-to-wheel energy consumption.

Long Answer Questions

1. Derive the longitudinal force balance for a vehicle climbing a road grade and explain each term.
2. Explain how wheel torque, motor torque, gear ratio, and drivetrain efficiency are related.
3. Describe the procedure for calculating energy consumption over a discrete drive cycle.
4. Discuss the physical and control limitations of regenerative braking.
5. Explain why continuous and peak power ratings must be treated separately when sizing an electric powertrain.

Analytical / Application Questions

1. A 1,500 kg EV with Cᵣ = 0.012, Cᴅ = 0.29, and frontal area 2.0 m² travels at 80 km/h on a level road. Using air density 1.225 kg/m³, calculate rolling resistance, aerodynamic drag, wheel power, and battery power for 90% propulsion efficiency.
2. Determine the approximate wheel force required for a 1,800 kg vehicle to climb a 10% grade at 50 km/h, neglecting acceleration. State the approximations used.
3. Compare the kinetic energy and ideal stopping distance of a vehicle at 60 km/h and 100 km/h for the same mass and constant deceleration. Explain the engineering implications for braking and regenerative-energy recovery.

Multiple Choice Questions with Answers

Question 1

Which force varies approximately with the square of relative air speed?

A. Rolling resistance

B. Aerodynamic drag

C. Grade force

D. Inertial force

Correct Answer: B. Aerodynamic drag

Question 2

A vehicle travels at 72 km/h. What is its speed in SI units?

A. 10 m/s

B. 15 m/s

C. 20 m/s

D. 25 m/s

Correct Answer: C. 20 m/s

Question 3

For a vehicle on level ground moving at constant speed, which term is zero in the longitudinal force balance?

A. Rolling resistance

B. Aerodynamic drag

C. Grade force

D. Inertial force

Correct Answer: D. Inertial force

Question 4

If vehicle speed doubles while mass and road conditions remain unchanged, kinetic energy becomes:

A. Half as large

B. Twice as large

C. Three times as large

D. Four times as large

Correct Answer: D. Four times as large

Question 5

A 1,000 kg vehicle experiences an acceleration of 2 m/s². Neglecting other forces, what inertial force is required?

A. 500 N

B. 1,000 N

C. 2,000 N

D. 20,000 N

Correct Answer: C. 2,000 N

Question 6

Which condition most directly limits regenerative braking when the battery is nearly full?

A. Low rolling resistance

B. Limited battery charge-power acceptance

C. Large frontal area

D. Low auxiliary load

Correct Answer: B. Limited battery charge-power acceptance

Question 7

Wheel power is calculated from tractive force and vehicle speed as:

A. P = F/v

B. P = F + v

C. P = Fv

D. P = mv²

Correct Answer: C. P = Fv

Question 8

A 90% efficient propulsion system delivers 18 kW at the wheels and has no auxiliaries. Approximate battery power is:

A. 16.2 kW

B. 18.0 kW

C. 20.0 kW

D. 21.8 kW

Correct Answer: C. 20.0 kW

Question 9

Which requirement is primarily associated with low-speed vehicle launch?

A. High motor torque

B. High aerodynamic drag coefficient

C. Low battery voltage only

D. Zero tyre-road friction

Correct Answer: A. High motor torque

Question 10

Why must a vehicle model specify whether grade is expressed as a percentage or an angle?

A. They are always identical

B. Percentage grade uses tan θ, whereas force uses sin θ

C. Angle is used only for rolling resistance

D. Percentage grade has units of newtons

Correct Answer: B. Percentage grade uses tan θ, whereas force uses sin θ