Vehicle Dynamics

This page explains the physical model behind BobDyn. A vehicle is a coupled, nonlinear dynamic system: states, constraints, force laws, compliance, actuation, and feedback act together. Bicycle models, roll centers, and linear tire stiffness are reduced views of that system. They stay useful when you know which assumptions they hide.

A metric name does not create a vehicle's response. Forces, moments, inertias, constraints, and time do. BobDyn keeps raw signals next to the metrics derived from them, so the physical connection stays visible.

The page assumes calculus, ODEs, and basic controls language. You do not need to derive the equations to use the ideas. For what an FSAE team can claim from simulation, tests, and telemetry, see FSAE Bridge.

Models and reductions

A high-fidelity multibody vehicle is a differential-algebraic equation (DAE), not a state-space model. Its algebraic part holds what an explicit ODE hides: constraints, reaction forces, contact conditions, tire force laws, and actuator relations. Motion and force are solved together, subject to the constraints.

A state-space model is a reduced form. It comes from projecting the constrained system onto independent coordinates, trimming it around an operating point, or linearizing it. Bicycle models, understeer gradients, yaw-rate gains, frequency response functions, and control-oriented handling metrics all live at this level. They are valid reduced views of the DAE when you remember what the reduction removed.

The DAE and state-space forms

The general form:

$$F(\dot{x},x,z,u,p,t)=0$$$$y=g(x,z,u,p,t)$$

$x$ holds dynamic states, $z$ holds algebraic variables, $u$ holds driver or actuator inputs, $p$ holds design parameters, and $y$ is measured behavior.

A constrained multibody system is normally written in second-order form:

$$M(q,p)\ddot{q}+{\mathrm{\Phi }}_q(q,p,t{)}^T\lambda =Q(q,\dot{q},u,p,t)$$$$\mathrm{\Phi }(q,p,t)=0$$

$q$ are generalized coordinates, $M$ is the mass matrix, $\mathrm{\Phi }$ are kinematic constraints, ${\mathrm{\Phi }}_q$ is the constraint Jacobian, $\lambda$ are constraint reaction multipliers, and $Q$ collects applied, inertial, tire, aero, spring, damper, actuator, and contact forces.

The reduced state-space form:

$${\dot{x}}_r=f_r(x_r,u,p,t)$$$$y=g_r(x_r,u,p,t)$$

Around a trim point, the local linear model is:

$$\delta {\dot{x}}_r=A\delta x_r+B\delta u$$$$\delta y=C\delta x_r+D\delta u$$

Geometry and response

Suspension geometry matters because it changes tire states, contact patch loads, force paths, motion ratios, and compliance. A diagram label matters only as far as it predicts those effects.

The design target is vehicle response:

• steady-state balance
• transient yaw response
• lateral acceleration buildup
• driver confidence
• contact patch load control
• tire force availability
• robustness to speed, ride height, load, and uncertainty

So ask "how does this system transmit forces and moments, and what response does that produce?", not "what is the roll center height?". Geometry shapes the dynamic system, but it is not all of it. Tie every geometry metric back to the loads, motions, and response it is meant to predict.

Roll centers

Roll centers are visualizations, not physical parts or force application points.

The classic construction uses pinned supports and two four-bar linkages. It is a planar approximation. It is good for fast reasoning about roll gradient, spring and damper deflection, and rough geometric load transfer.

Instant centers are more fundamental within that approximation. They describe a linkage's instantaneous motion. For one corner, you can approximate load transmission from unsprung to sprung mass as a virtual link from the contact patch toward an instant center. In a full spatial view, the corner has an instantaneous screw axis: it rotates about and translates along a virtual axis. That model is usually more useful for design than roll center height alone.

The strongest quantification is force-based. Apply a force, then measure the support reaction, the jacking response, and the change in contact patch load. That is the response the vehicle sees. The roll center is a model coordinate that tries to summarize it.

Jacking and anti-geometry

Anti-dive, anti-squat, and anti-roll describe how geometry changes load transmission between unsprung and sprung mass. The question is: when a force enters the tire contact patch, how much of it creates a vertical reaction on the frame?

Draw the front-view and side-view instant centers and connect them. That line is the virtual axis for the corner. The corner has both an instantaneous motion structure and a force transmission structure. It does not "push through a roll center".

Anti-geometry is geometric resistance to sprung-mass attitude change. A useful reference is the jacking force that would fully resist the attitude moment. For a longitudinal case:

$$F_{\text{jack, 100\%}}=F_x\frac{h_{\text{cg}}}{L}$$

For a lateral case:

$$F_{\text{jack, 100\%}}=F_y\frac{h_{\text{cg}}}{T}$$

$h_{\text{cg}}$ is CG height, $L$ is wheelbase, and $T$ is track width. 100% anti-dive means the suspension load path matches the jacking response needed to resist the attitude moment. It is not a special point in space. The exact value still depends on force distribution, sign convention, suspension force lines, and the case under analysis. An instant center that points "the wrong way" compared to a textbook diagram does not break the physics. The diagram was a special case.

FourPostEval reports anti percentages on this scale. See Suspension, kinematics, and compliance.

Load transfer

Load transfer starts from force and moment balance for the whole vehicle:

$$\sum F=m{a}_G$$$$\sum M_G=I_G\alpha +\omega \times I_G\omega$$

Springs, bars, dampers, geometry, and chassis stiffness change how the required loads are distributed and how fast they appear. They do not remove the balance requirement.

In lateral steady-state analysis, lateral load transfer distribution is:

$$LLTD=\frac{\mathrm{\Delta }{F}_{z,\text{front}}}{\mathrm{\Delta }{F}_{z,\text{front}}+\mathrm{\Delta }{F}_{z,\text{rear}}}$$

Mass, CG height, track width, and lateral acceleration set most of the total lateral load transfer. Springs and bars mainly shift its front/rear distribution, not the total.

For the fuller picture, model the chassis as three torsional springs in series: front roll stiffness, chassis torsional stiffness, and rear roll stiffness. If the chassis twists, front and rear no longer see the same roll input. The nominal rigid-frame LLTD then becomes harder to achieve and more dynamic. In a very stiff chassis, front/rear roll stiffness sets the elastic distribution. As torsional stiffness drops, the axles decouple and the car drifts from the intended rigid-frame behavior.

One check is whether achievable LLTD stays within a tolerance band of the nominal target. A 1% band is a common first cutoff. The more reliable check is dynamic: change torsional rigidity, run the response, and compare the steady-state and transient metrics you care about.

Damping

Springs and bars shape where load transfer settles. Dampers shape how fast it gets there. Dampers set how quickly tire loads build, how quickly yaw moment appears, how much contact patch load overshoot occurs, and how the car feels during turn-in, release, braking, and combined maneuvers.

In simple form:

$$F_d=c{v}_d$$

$F_d$ is damper force, $c$ is the damping coefficient, and $v_d$ is damper velocity. Real dampers are not linear, but the core behavior holds: force reacts to velocity, not displacement.

So damping is a transient load-transfer tool as well as a ride tool. More front damping can make front tire loads build sooner and shift the early yaw moment. More rear damping can settle the rear faster, or resist motion in ways that change phase and driver confidence.

Excessive high-speed compression force hurts sprung-mass NVH and adds contact patch load variation. That variation costs grip because tires are load-sensitive. Added normal load does not buy proportional force capacity, so oscillating normal load usually wastes grip.

Tires

A tire is a nonlinear force law, not a scalar friction coefficient. Its force depends on normal load, slip angle, slip ratio, camber, temperature, pressure, history, and wear. It also builds force with a lag and moves its effective point of force application. For the full explanation, see Tire Behavior.

Aero

Aero is platform-sensitive force generation. Speed-squared scaling is the easy part:

$$q=\frac{1}{2}\rho V^2$$$$F_{\text{aero}}=qSC(\text{platform})$$

The hard part is $C(\text{platform})$. For a race car, platform can include corner ride heights, pitch, roll, yaw, body slip, roadwheel angle, wheel wake, ground proximity, and upstream boundary conditions. Downstream flow control often decides whether a package works. Geometric inaccuracy, surface quality, mounting error, and boundary-condition mismatch are all real sources of uncertainty.

A common approach is to compute steady-state CFD forces across a parameterized attitude space, then interpolate an aero map:

$$F_{\text{aero}}=F(V,h_{\text{FL}},h_{\text{FR}},h_{\text{RL}},h_{\text{RR}},\beta ,\delta ,...)$$

This works with enough compute and a meaningful parameterization. The map is only as good as its coverage, input fidelity, and validation.

Platform control matters because aero balance moves with ride height and pitch. With speed, the car can gain front-biased or rear-biased downforce, drag, pitch moment, or roll and yaw moment. All of these change tire loads and dynamic response. Transient aero is harder: a platform change does not always produce an instant force response. A fuller model needs force-generation time constants or dynamics from transient CFD. Aero is a force law coupled to the suspension platform, not a single coefficient.

Torsional rigidity

Treat chassis torsional rigidity as coupled compliance, not a target number. The question is whether chassis compliance changes the response you want to control.

A rigid-frame model assumes front and rear suspension share one body motion. A compliant chassis weakens that. The axles roll more independently, and the actual load transfer distribution drifts from the rigid-frame value. The transient response also changes, because the chassis adds a compliance path and stores energy.

Judge torsional rigidity by its effects:

• LLTD error from nominal
• available LLTD adjustment range
• roll gradient
• yaw response
• lateral acceleration response
• contact patch load variation
• frequency response
• driver confidence

Production vehicles sometimes use compliance on purpose, in bushings, steering, subframes, tires, and structure. It can filter noise, improve robustness, shape feel, or protect components. In most FSAE cars, the first goal is to minimize uncontrolled compliance, so the car does what the engineer expects. Compliance is not bad in itself. Unmodeled compliance is.

Suspension

The suspension serves the tire. It is a passive mechanical system that keeps the tires in useful operating states across the car's range of motion and loading.

Suspension design changes these properties, and each one changes force generation, moment generation, or time response:

Property	Why it matters
Camber	Tires are camber-sensitive.
Toe	Creates slip angle and yaw moment.
Caster, KPI, mechanical trail, scrub radius	Contact-patch forces create moments about the steering and suspension axes.
Motion ratio, spring and damper velocity	A component-level spring or damper rate is not the wheel-level rate.

Jacking response, anti behavior, roll stiffness distribution, and contact patch load variation also change force generation, moment generation, or time response.

Kinematics and compliance testing is system identification. It lets an engineer compare the design to the physical car. Analytical calculations can get close. Physical compliance changes the effective behavior: wheel bearings, joints, tires, and structural interfaces with unloaded, seated, and snubbed rates. Ask the downstream question: what did this do to the tire, and what did the vehicle do in response?

Transient response

Driver confidence is a dynamic systems problem. A driver does not feel a roll center. They feel the time history of yaw rate, lateral acceleration, roll, steering torque, sideslip, and tire capacity. Frequency response, phase, lag, damping, and bandwidth describe how the car communicates with the driver.

For yaw response:

$$G_r(j\omega )=\frac{R(j\omega )}{{\mathrm{\Delta }}_{\text{HWA}}(j\omega )}$$

For lateral acceleration response:

$$G_{a_y}(j\omega )=\frac{A_y(j\omega )}{{\mathrm{\Delta }}_{\text{HWA}}(j\omega )}$$

Magnitude says how much response the car produces. Phase says when it arrives.

For many driver-confidence targets, useful yaw-rate response should begin before full lateral acceleration builds. The car must rotate toward a stable yaw state before the full lateral demand arrives. If yaw develops too slowly and lateral acceleration arrives suddenly, two things go wrong. The driver gets a poor on-center feel. The tires take a sharp contact patch load event, which costs grip in a nonlinear tire.

Open-loop tests expose the plant directly: ramp steer, step steer, and frequency response. They show quasi-steady behavior, transient buildup, overshoot, delay, and phase, with no driver feedback loop to hide them. Closed-loop driver behavior is the final test. Open-loop testing shows what the vehicle is before a driver has to control it.

Understeer gradient

Understeer gradient is a local slope. In the linear region (often about 0.1 g to 0.4 g in practical evaluations):

$$K={\frac{\partial {\delta }_{\text{excess}}}{\partial a_y}|}_{\text{linear}}$$

${\delta }_{\text{excess}}$ is the steering input above the geometric curvature requirement. The gradient shows how required steering changes with lateral acceleration in a local region. One slope cannot describe the whole vehicle. A car can have a reasonable linear understeer gradient and still be poor in transient response, limit behavior, contact patch load control, or driver confidence. Metrics are measurements, not complete explanations.

BobSim reports linear and limit understeer gradients from steady-state sweeps. See Steady-state handling.

Why simulation matters

The physical system is too coupled to judge from isolated metrics. A change to spring rate, damper curve, tire pressure, aero platform, geometry, or chassis stiffness rarely affects one behavior. It propagates through loads, states, constraints, and force laws.

Simulation makes those connections repeatable. The same vehicle definition runs through the same maneuvers, with the same signal definitions, fitting methods, and output metrics. That does not make the model correct. It makes the assumptions inspectable and the results comparable. This is why BobDyn reports keep both the trace and the summary. A steady-state sweep may report understeer gradient, but the steering, curvature, roll, sideslip, and acceleration traces still matter.

Standard tests reduce the vehicle to measurable outputs: ramp steer, step steer, frequency response, K&C-style sweeps, and envelope studies. They give a common language for correlation, debugging, and design exploration. A K&C sweep identifies geometry and compliance. A maneuver simulation shows the response those properties produce. Use them together.

The design questions

Vehicle dynamics becomes clear when you force every named concept back into the dynamic system:

• What are the states?
• What are the inputs?
• What are the force laws?
• What are the constraints?
• What stores energy?
• What dissipates energy?
• What is measured?
• What response is desired?

Roll centers, LLTD, cornering stiffness, aero balance, damping ratio, natural frequency, understeer gradient, and bandwidth compress behavior into engineering language. The vehicle responds to the physical system underneath, not to the language. The work of vehicle dynamics is to make the physics produce the behavior you want.