m physical modeling / IICSM

PHYSICS / ENGINEERING / COMPUTING

Car design: road loads, handling, ride and energy

Relate vehicle mass, tires, aerodynamics, gearing and suspension to longitudinal performance and basic handling.

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1. Longitudinal force balance

Definitions & inputs. m mass,a acceleration,V speed,ρ air density,CdA drag area,Crr rolling coefficient,θ road angle.

  1. Aerodynamic drag and rolling loss oppose motion.

    FD=12ρCDAV2,FR=Crrmgcos⁡θF_D=\tfrac12\rho C_DAV^2,\quad F_R=C_{rr}mg\cos\theta

  2. Newton’s law combines acceleration and road loads.

    Fwheel=ma+FD+FR+mgsin⁡θF_{wheel}=ma+F_D+F_R+mg\sin\theta

  3. Convert wheel force to mechanical power at the road.

    Pwheel=FwheelVP_{wheel}=F_{wheel}V

Interpretation. Rotational inertia, speed-dependent tire losses and accessories add terms to an energy-consumption model.

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2. Gearing and traction

Definitions & inputs. Tm motor/engine torque,i ratio,η drive line efficiency,rw wheel radius,μ tire friction.

  1. Mechanical gearing trades shaft speed for wheel torque.

    Tw=ηiTm,Fw=Tw/rwT_w=\eta iT_m,\quad F_w=T_w/r_w

  2. Kinematic speed compatibility connects road speed to shaft speed.

    ωm=iV/rw\omega_m=iV/r_w

  3. Driven-axle normal load limits usable traction.

    ∣Fw∣≤μNdriven|F_w|\le\mu N_{driven}

Interpretation. Torque and power limits, tire slip and load transfer must be applied together.

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3. Braking and cornering

Definitions & inputs. a deceleration magnitude,R curve radius,L wheelbase,h CG height.

  1. Integrate constant deceleration; perception and reaction distance are separate.

    d=V02/(2a),a≤μgd=V_0^2/(2a),\quad a\le\mu g

  2. Centripetal acceleration and low-speed steering geometry set separate handling scales.

    ay=V2/R,δ≃L/Ra_y=V^2/R,\quad\delta\simeq L/R

  3. Braking transfers axle load through the CG height.

    ΔN=mah/L\Delta N=mah/L

Interpretation. Simultaneous braking and cornering share a tire-friction budget; this is not a prediction of tire behavior at the limit.

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4. Ride and thermal energy

Definitions & inputs. k suspension stiffness,c damping,m sprung quarter-car mass,ζ damping ratio.

  1. A spring-damper model relates force to vertical displacement.

    mz¨+cz˙+kz=F(t)m\ddot z+c\dot z+kz=F(t)

  2. Normalize the ODE to identify frequency and damping.

    ωn=k/m,ζ=c/(2km)\omega_n=\sqrt{k/m},\quad\zeta=c/(2\sqrt{km})

  3. Stopping energy must go into regeneration, drag and heat.

    Estop=12mV2E_{stop}=\tfrac12mV^2

Interpretation. Full vehicle design couples ergonomics, packaging, structure, powertrain, sensors, control and safety; these equations provide subsystem checks.

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Graphical worked example

No-wind drag and rolling resistance for the stated 1500 kg vehicle; no drivetrain or accessory losses. X axis: Road speed (m/s). Y axis: Level-road wheel power (kW).
No-wind drag and rolling resistance for the stated 1500 kg vehicle; no drivetrain or accessory losses. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Aerodynamic drag

Definitions & inputs. ρ1.2,Cd.3,A2.2m²,V30m/s.

  1. Choose the governing model and isolate the requested quantity.

    D=ρCDAV2/2D=\rho C_DAV^2/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5(1.2)(.3)(2.2)(302).5(1.2)(.3)(2.2)(30^2)

  3. Evaluate the expression; the result uses the units shown.

    Result=356.4 N\mathrm{Result}=356.4\ \mathrm N

Interpretation. No wind.

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Example 02. Rolling resistance

Definitions & inputs. Crr.01,m1500kg,g9.81.

  1. Choose the governing model and isolate the requested quantity.

    R=CrrmgR=C_{rr}mg

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .01(1500)(9.81).01(1500)(9.81)

  3. Evaluate the expression; the result uses the units shown.

    Result=147.15 N\mathrm{Result}=147.15\ \mathrm N

Interpretation. Level road.

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Example 03. Cruise wheel power

Definitions & inputs. Drag356.4N,rolling147.15N,V30m/s.

  1. Choose the governing model and isolate the requested quantity.

    P=(D+R)VP=(D+R)V

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (356.4+147.15)(30)(356.4+147.15)(30)

  3. Evaluate the expression; the result uses the units shown.

    Result=15106.5 W\mathrm{Result}=15106.5\ \mathrm W

Interpretation. Excludes accessories and drivetrain loss.

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Example 04. Grade force

Definitions & inputs. m1500kg,sinθ.05.

  1. Choose the governing model and isolate the requested quantity.

    Fg=mgsin⁡θF_g=mg\sin\theta

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1500(9.81)(.05)1500(9.81)(.05)

  3. Evaluate the expression; the result uses the units shown.

    Result=735.75 N\mathrm{Result}=735.75\ \mathrm N

Interpretation. Stated sinθ,not exact percent grade conversion.

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Example 05. Acceleration force

Definitions & inputs. m1500kg,a2m/s².

  1. Choose the governing model and isolate the requested quantity.

    F=maF=ma

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1500(2)1500(2)

  3. Evaluate the expression; the result uses the units shown.

    Result=3000 N\mathrm{Result}=3000\ \mathrm N

Interpretation. Add road loads for required wheel force.

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Example 06. Wheel torque

Definitions & inputs. F3000N,r.3m.

  1. Choose the governing model and isolate the requested quantity.

    T=FrT=Fr

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    3000(.3)3000(.3)

  3. Evaluate the expression; the result uses the units shown.

    Result=900 N m\mathrm{Result}=900\ \mathrm{N\,m}

Interpretation. Total drive wheel torque.

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Example 07. Motor torque

Definitions & inputs. Tw900Nm,i9,η.95.

  1. Choose the governing model and isolate the requested quantity.

    Tm=Tw/(iη)T_m=T_w/(i\eta)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    900/(9(.95))900/(9(.95))

  3. Evaluate the expression; the result uses the units shown.

    Result=105.2632 N m\mathrm{Result}=105.2632\ \mathrm{N\,m}

Interpretation. Fixed gearing.

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Example 08. Motor speed

Definitions & inputs. V30m/s,r.3m,i9.

  1. Choose the governing model and isolate the requested quantity.

    ωm=iV/r\omega_m=iV/r

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    9(30)/.39(30)/.3

  3. Evaluate the expression; the result uses the units shown.

    Result=900 rad/s\mathrm{Result}=900\ \mathrm{rad/s}

Interpretation. Slip neglected.

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Example 09. Shaft rpm

Definitions & inputs. ω900rad/s.

  1. Choose the governing model and isolate the requested quantity.

    n=60ω/(2π)n=60\omega/(2\pi)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    60(900)/(2π)60(900)/(2\pi)

  3. Evaluate the expression; the result uses the units shown.

    Result=8594.367 rpm\mathrm{Result}=8594.367\ \mathrm{rpm}

Interpretation. Angular-speed conversion.

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Example 10. Adhesion force

Definitions & inputs. μ.8,Ndriven7000N.

  1. Choose the governing model and isolate the requested quantity.

    Fmax=μNF_{max}=\mu N

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .8(7000).8(7000)

  3. Evaluate the expression; the result uses the units shown.

    Result=5600 N\mathrm{Result}=5600\ \mathrm N

Interpretation. Ideal friction ceiling.

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Example 11. Braking distance

Definitions & inputs. V20m/s,a7m/s².

  1. Choose the governing model and isolate the requested quantity.

    d=V2/(2a)d=V^2/(2a)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    400/14400/14

  3. Evaluate the expression; the result uses the units shown.

    Result=28.57143 m\mathrm{Result}=28.57143\ \mathrm m

Interpretation. Braking only,excludes reaction.

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Example 12. Reaction distance

Definitions & inputs. V20m/s,t1s.

  1. Choose the governing model and isolate the requested quantity.

    d=Vtd=Vt

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    20(1)20(1)

  3. Evaluate the expression; the result uses the units shown.

    Result=20 m\mathrm{Result}=20\ \mathrm m

Interpretation. Add to braking distance for this assumed reaction time.

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Example 13. Cornering acceleration

Definitions & inputs. V15m/s,R50m.

  1. Choose the governing model and isolate the requested quantity.

    ay=V2/Ra_y=V^2/R

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    225/50225/50

  3. Evaluate the expression; the result uses the units shown.

    Result=4.5 m/s2\mathrm{Result}=4.5\ \mathrm{m/s^2}

Interpretation. Steady turn.

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Example 14. Low-speed steering

Definitions & inputs. L2.7m,R50m.

  1. Choose the governing model and isolate the requested quantity.

    δ≃L/R\delta\simeq L/R

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2.7/502.7/50

  3. Evaluate the expression; the result uses the units shown.

    Result=0.054 rad\mathrm{Result}=0.054\ \mathrm{rad}

Interpretation. Bicycle approximation.

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Example 15. Load transfer

Definitions & inputs. m1500,a7,h.5,L2.7SI.

  1. Choose the governing model and isolate the requested quantity.

    ΔN=mah/L\Delta N=mah/L

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1500(7)(.5)/2.71500(7)(.5)/2.7

  3. Evaluate the expression; the result uses the units shown.

    Result=1944.444 N\mathrm{Result}=1944.444\ \mathrm N

Interpretation. Axle load increment on level road.

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Example 16. Quarter-car frequency

Definitions & inputs. k20000N/m,m350kg.

  1. Choose the governing model and isolate the requested quantity.

    fn=k/m/(2π)f_n=\sqrt{k/m}/(2\pi)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    20000/350/(2π)\sqrt{20000/350}/(2\pi)

  3. Evaluate the expression; the result uses the units shown.

    Result=1.203098 Hz\mathrm{Result}=1.203098\ \mathrm{Hz}

Interpretation. Single ride mode.

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Example 17. Critical damping

Definitions & inputs. Samek,m.

  1. Choose the governing model and isolate the requested quantity.

    cc=2kmc_c=2\sqrt{km}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    220000(350)2\sqrt{20000(350)}

  3. Evaluate the expression; the result uses the units shown.

    Result=5291.503 N s/m\mathrm{Result}=5291.503\ \mathrm{N\,s/m}

Interpretation. Critical damping coefficient.

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Example 18. Damping ratio

Definitions & inputs. c1500,k20000,m350SI.

  1. Choose the governing model and isolate the requested quantity.

    ζ=c/(2km)\zeta=c/(2\sqrt{km})

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1500/[220000(350)]1500/[2\sqrt{20000(350)}]

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2834734 \mathrm{Result}=0.2834734\ {}

Interpretation. Linear ride model.

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Example 19. Stopping energy

Definitions & inputs. m1500kg,V20m/s.

  1. Choose the governing model and isolate the requested quantity.

    E=mV2/2E=mV^2/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5(1500)(400).5(1500)(400)

  3. Evaluate the expression; the result uses the units shown.

    Result=300000 J\mathrm{Result}=300000\ \mathrm J

Interpretation. Rotating inertia omitted.

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Example 20. Cruise energy per distance

Definitions & inputs. Constant wheel power15kW,speed100km/h.

  1. Choose the governing model and isolate the requested quantity.

    e=P/Ve=P/V

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    15/10015/100

  3. Evaluate the expression; the result uses the units shown.

    Result=0.15 kWh/km\mathrm{Result}=0.15\ \mathrm{kWh/km}

Interpretation. Wheel energy only.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.