Build the bicycle model from axle side-force balance

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Course: Read the forces that steer the car

Module: Balance the car with forces and moments

Estimated duration: 45 minutes

The bicycle model is the simplest useful way to turn a whole car into a force-balance problem. You replace the two front tires with one equivalent front wheel on the centerline, replace the two rear tires with one equivalent rear wheel, put the center of mass between them, and ask a precise question: what front and rear side forces are required for this car to follow this curved path without building an unbalanced yaw moment?

That last phrase is the lesson. The model is not valuable because it looks like a bicycle. Dixon is explicit that the name comes from compressing the pairs of real wheels into single centerline wheels, not because the handling is related to a real leaning bicycle. Its value is that it makes the balance visible. The car corners because the front and rear axles generate side force. The car holds a steady attitude because those side forces also balance their moments about the center of mass. If the lateral force sum is right but the moment balance is wrong, the car does not have steady-state cornering. It has a yaw acceleration or a changing yaw rate.

In this lesson you will build that model from the axle side-force balance, not from a memorized final equation. You will learn what each term means physically, how the front and rear axle forces enter the model, why the center-of-mass distances matter, and how to read the result as understeer, oversteer, static stability, and yaw damping without jumping ahead into the separate understeer-gradient calculation lesson.

Start with the picture in your head. Draw the car from above. Put the center of mass at G. The front axle is a distance a ahead of G. The rear axle is a distance b behind G. The wheelbase is L, so L equals a plus b. The front axle can be steered by a small angle. The car has a body sideslip or yaw angle beta relative to its path direction. In steady cornering it also has a yaw rate. The tires do not point exactly along their paths; they operate at slip angles. In the linear range, a tire side force is proportional to slip angle through cornering stiffness. Gillespie states the key low-slip simplification: at low slip angles, about 5 degrees or less, the relationship between cornering force and slip angle is treated as linear. That is what lets the bicycle model become algebra instead of a full tire model.

The front axle is not one tire in the real car, so its cornering stiffness is the sum for the two front tires in the simplified model. Dixon gives the front-axle stiffness as two times the single-front-wheel stiffness. The same compression is made at the rear. This means the front axle side force is the front axle cornering stiffness multiplied by the front slip angle, with sign convention handled consistently; the rear axle side force is the rear axle cornering stiffness multiplied by the rear slip angle. For this lesson, keep the physical meaning higher than the sign convention: more slip angle at an axle requires more side force at that axle while the tire is still in the linear range.

Now write the first balance. In a steady turn, the tire side forces must provide the centripetal acceleration. Gillespie gives the force statement directly: the sum of the front and rear lateral forces equals mass times speed squared divided by turn radius. In model language, Fyf plus Fyr equals M V squared over R. That is the axle side-force balance. It answers how much total lateral force the car needs.

But that balance alone does not tell you whether the car is in a steady attitude. You also need the yaw moment balance about G. The front side force acts at a lever arm a ahead of G. The rear side force acts at a lever arm b behind G. If the front and rear side forces do not create equal and opposite yawing moments, the vehicle will not remain in a constant yaw state. Dixon describes steady state as the condition where steering gives a suitable front slip angle to balance the rear-wheel slip angle and produce zero yaw moment, and therefore constant yaw speed. That is the second balance: the front axle moment and rear axle moment must cancel.

This is the first major habit to build. Do not think of the bicycle model as a steering-angle formula. Think of it as two balances. The lateral force balance sets the required total side force. The yaw moment balance decides how that required side force must be distributed between the front and rear axles. Once you know the force distribution, the tire cornering stiffnesses tell you what slip angles must exist. Once you know the slip angles, the vehicle geometry tells you what steering angle and body sideslip are compatible with that turn.

For intermediate driving analysis, that order matters. If you jump straight to steering angle, you hide the mechanism. The car does not understeer because a formula says the steering angle is larger. It understeers because the front axle must operate at a larger slip angle than the rear axle for the same steady turn, so the driver must add steering beyond the kinematic angle. The car does not oversteer because the rear feels loose in a vague sense. It oversteers when the rear axle side-force requirement and rear tire capability demand the larger rear slip angle, so the yaw moment balance is achieved with a different steering relationship. The formulas are summaries of that axle force story.

Build the model in six steps.

Step one: compress the car to two axles on the centerline. The two real front tires become one equivalent front axle side force. The two real rear tires become one equivalent rear axle side force. Put the front axle at plus a from G and the rear axle at minus b from G. This is why the wheelbase position of the center of mass matters. A car with the center of mass closer to the front axle gives the rear force a longer moment arm and the front force a shorter one. A car with the center of mass closer to the rear axle does the opposite. The axle forces are not just forces; they are forces applied at lever arms.

Step two: state the assumptions before using the model. You are in steady-state cornering, not an entry transient. The turn radius is large compared with the wheelbase, so the small-angle simplification is acceptable. The tire slip angles are low enough for the linear cornering-stiffness approximation. Aerodynamic forces, rolling resistance, suspension compliance, and tractive-force complications are either neglected or treated outside this first model. Dixon repeatedly shows the simple model in this stripped condition, while also warning that suspension neglect is a serious practical limitation when applying the coefficients to a complete vehicle. That warning is part of the model, not an afterthought.

Step three: write the lateral force balance. The total side force needed is M V squared over R. The axles provide it: Fyf plus Fyr. This is the cleanest entry point for the model because it tells you that speed is expensive. Double speed at the same radius and the required side force goes up by four. That point should sound familiar from vehicle dynamics, but here you keep it tied to the two axles: the increase must be carried by the front side force, the rear side force, or both.

Step four: write the yaw moment balance. The front axle side force has a moment arm a. The rear axle side force has a moment arm b. For steady-state cornering, the net yaw moment about G is zero. In the simplest sign convention, the front contribution and rear contribution oppose each other, so the magnitudes satisfy a times Fyf equals b times Fyr. This does not mean the front and rear forces are equal. It means the force at the short lever arm must be larger than the force at the long lever arm to produce the same moment.

Step five: solve the two balances together. From the moment balance, Fyf over Fyr equals b over a. From the force balance, Fyf plus Fyr equals M V squared over R. The front and rear side-force shares are therefore set by the center-of-mass position in the neutral linear model: the front axle carries the fraction b over L of the required lateral force, and the rear axle carries the fraction a over L. That is the simplest axle force split. It is the force split required for zero yaw moment in steady cornering before you add the extra steering and compliance effects that produce understeer gradient.

Step six: translate force requirement into slip angle requirement. Each axle has a cornering stiffness. A stiffer axle needs less slip angle to generate a given side force in the linear range. A softer axle needs more slip angle. This is where handling balance enters. The force split is set by mass, speed, radius, and center-of-mass location. The slip angle split is set by those force requirements divided by the front and rear axle cornering stiffnesses. If the front axle needs more slip angle than the rear, the driver must steer beyond the pure kinematic path angle; that is the understeer-angle idea Dixon introduces in the basic handling curve. If the rear needs more slip angle in the relevant sense, the rear contributes more strongly to yaw response and the vehicle can move toward oversteer.

Notice what you have deliberately not done. You have not calculated the understeer gradient in full. That is the sibling lesson. You have not mapped yaw rate and sideslip to steering and speed in detail. That is also a sibling lesson. Here you have built the skeleton that those lessons stand on: total side force, axle moment balance, axle cornering stiffness, and the center-of-mass lever arms.

Now deepen the model by separating three related but different conditions: steer input, body yaw angle, and yaw rate.

Dixon shows a vehicle with steer angle only and no body yaw or yaw rate. That condition creates a front slip angle, therefore a front lateral force, and therefore a yaw moment. It is not a steady-cornering solution by itself. It is a component case that helps you understand what steering contributes. Steering the front axle creates front side force, and because that force acts ahead of G it also creates a yawing moment.

Dixon also shows a vehicle with body yaw angle beta, no steer angle, and zero angular velocity. In that condition, the front and rear slip angles are equal to the yaw angle. Both axles develop side force. The total lateral force acts close to the center of mass, usually a little behind it. If that total force acts behind G, the resulting moment is restoring for a small yaw disturbance. This is where static margin enters the picture. Static margin is the moment arm of the total tire force behind G, often expressed as a fraction of wheelbase. If the force acts behind G, the static margin is positive and the vehicle has directional yaw static stability.

The physical statement is more useful than the label. A small yaw angle exposes both axles to slip angle. The front axle force tends to disturb the yaw angle in one direction; the rear axle force tends to restore it in the other. Dixon states the requirement physically: for static stability, the restoring moment from the rear axle must exceed the disturbing moment from the front axle. That condition depends on both axle stiffness and the lever arms a and b. A stiff rear axle far behind G is a strong stabilizing influence. A stiff front axle far ahead of G is a strong disturbing influence for this static yaw case.

This static stability idea is easy to confuse with steady cornering balance, so keep them separate. Static margin asks what happens when the vehicle is given a small yaw angle with no steering and no yaw rate. The steady cornering balances ask how front and rear side forces must combine to produce centripetal force with zero yaw acceleration. They use the same ingredients, but they answer different questions. One is a disturbance-response question. The other is a path-following equilibrium question.

Dixon then adds yaw speed. With a yaw rate, the front and rear axle velocities differ because one axle is ahead of G and one is behind it. The slip-angle increments are proportional to a times yaw rate over speed at the front and b times yaw rate over speed at the rear. The side forces generated by those slip-angle increments create a moment that opposes the yawing velocity. Dixon calls this yaw damping and notes that it is inversely proportional to speed while always opposing yaw velocity. In driver language, yaw damping is one reason the car resists continuing to rotate forever after a yaw-rate disturbance. At higher speed, this damping contribution from the same geometry is reduced by the speed term, which is one reason steady-state and transient balance need careful speed context.

These three component cases are the clean way to build intuition without memorizing a dense equation. Steering input supplies a front-force and front-moment contribution. Body yaw angle supplies front and rear slip-angle contributions and reveals the static margin. Yaw rate supplies front and rear slip-angle contributions that oppose yaw velocity and produce yaw damping. The complete linear bicycle model is the sum of those effects. That is why Dixon says the complete vehicle characteristics can be expressed through moments of the tire coefficients about the center of mass: a zeroth moment for total side-force stiffness, a first moment for yaw side-force or static-margin behavior, and a second moment for yaw damping behavior.

Those complete vehicle cornering stiffnesses are not extra magic terms. They are bookkeeping. The zeroth moment adds the axle cornering stiffnesses and tells you how strongly the whole vehicle generates side force from a common slip-angle-like deflection. The first moment weights those stiffnesses by distance from G and tells you whether the resultant force tends to act ahead of or behind G. The second moment weights the stiffnesses by distance squared and appears in yaw damping because yaw rate creates slip-angle increments that grow with lever arm. Dixon gives representative medium-car values for these three stiffness measures, but the lesson is the structure: add stiffness for force, weight stiffness by position for moment, and weight stiffness by position squared for damping.

You can now read the model like an instructor reads a car in a skidpad exercise. If the driver increases speed on the same radius, the required total side force rises. If the front axle has to take a larger share or has lower effective stiffness, its slip angle rises more. The driver sees that as more steering required for the same radius. If the rear axle loses effective cornering force because of load transfer, suspension geometry, braking, power, or tire saturation, the yaw moment balance changes and the car may rotate more than expected. Gillespie emphasizes that many vehicle-design factors influence cornering forces under lateral acceleration, including suspension and steering-system effects; the tire cornering stiffness derivation is the basis, not the whole car.

That is the bridge from theory to driving. The bicycle model does not tell you that every understeer complaint is only a front-tire problem. It tells you to ask which axle force changed, which lever arm it acts through, and whether the resulting side-force and moment balances still match the path. A front tire that requires more slip angle for the same side force produces understeer behavior. A rear axle that cannot maintain the restoring contribution needed for balance produces oversteer behavior. Load transfer and suspension effects can move either axle in those directions, which is why the simple tire-stiffness model illuminates the mechanism but does not replace the full vehicle.

There is one more boundary to respect. Dixon includes diagrams for rear-drive and front-drive vehicles with tractive forces added to maintain steady tangential speed. If the model neglects tractive force, the resultant tire force can contain a drag component and the tangential speed is not exactly constant. With rear drive or front drive, the total tire force direction changes because longitudinal force is present at the driven axle. For the first bicycle-model lesson, you can neglect that complication when the problem is pure steady-state side-force balance. But on track, braking and throttle alter tire force direction and available lateral force. That belongs in more complete combined-force and transient lessons, but you should not forget that the pure side-force bicycle model is a stripped-down tool.

A useful mental test is to build the model from scratch on a blank page in under two minutes. Draw G. Mark a forward distance a to the front axle and a rearward distance b to the rear axle. Write L equals a plus b. Draw Fyf at the front axle and Fyr at the rear axle. Write Fyf plus Fyr equals M V squared over R. Then write the zero-yaw-moment condition about G. Finally write front slip angle produces front force through front axle cornering stiffness, and rear slip angle produces rear force through rear axle cornering stiffness. If you can do that without looking up the final understeer-gradient equation, you own the model at the level this lesson requires.

The payoff is diagnostic. When you look at a data trace or listen to a driver, you can sort comments into the model. More steering needed at the same radius and speed points to front slip angle increasing relative to the path requirement. A car that takes a yaw set and then keeps rotating points to insufficient rear restoring contribution or too much front disturbing contribution in that state. A car that feels slow to build yaw rate may have strong yaw damping or insufficient front moment from steering input. These are not final diagnoses by themselves, but they are the correct first questions because they come directly from axle side-force and moment balance.

The model is also a guardrail against a common beginner mistake in vehicle dynamics: treating lateral acceleration as one lump at the center of mass and forgetting the axles. The center of mass is where you write the inertial requirement, but the tires are where the forces are generated. The distance from each tire force to G determines whether a given side force stabilizes, destabilizes, or balances yaw. Every serious handling calculation in this module is a more detailed version of that idea.

Use this lesson as the foundation for the next two skills. When you calculate understeer gradient from tire and geometry data, you are quantifying how the required slip-angle difference changes with lateral acceleration. When you map yaw rate and sideslip to steering and speed, you are taking the steering, yaw angle, and yaw-rate component cases and solving them together. The bicycle model you built here is the common language under both tasks.

Worked example: neutral steady corner on a constant-radius skidpad

Imagine a car held on a constant-radius skidpad at steady speed. You are not analyzing turn-in, brake release, throttle application, or tire saturation. You are asking for the simplest steady condition: what front and rear side forces must exist so the car follows the circle without gaining yaw acceleration?

Draw the bicycle model. G sits between the axles. The front axle is a ahead of G. The rear axle is b behind G. The total required lateral force is M V squared over R. That force must be supplied by Fyf plus Fyr. Now enforce zero moment about G. The front side force acts through a. The rear side force acts through b. The moment balance requires the two axle moments to cancel.

If a is shorter than b, the front axle has the shorter lever arm and must carry the larger side force to make the same moment. If b is shorter than a, the rear axle has the shorter lever arm and must carry the larger side force. Nothing about this step requires you to know understeer gradient. You are simply distributing the required side force so the moments cancel.

Now convert force to slip angle. If the front axle cornering stiffness is high, the front can generate its required side force with a smaller front slip angle. If the rear axle cornering stiffness is low, the rear needs a larger rear slip angle for its share. That slip-angle comparison is where the driver-facing balance begins. The steering wheel position you see from the cockpit is downstream of this force and stiffness story.

The success criterion for this worked example is not a number. It is whether you can explain why equal front and rear side force is usually not the correct assumption. Equal moment is the condition for steady yaw balance, and the lever arms decide what force split gives that moment balance.

Worked example: small yaw disturbance and static margin

Now take the same bicycle model but remove steering and yaw rate. Give the vehicle a small body yaw angle beta. Dixon describes this as the case where the front and rear slip angles are equal to the yaw angle. Both axles develop side force because both axles are now moving at a small angle to where their wheels point.

The question changes. You are no longer asking what force split follows a chosen radius. You are asking whether the tire-force resultant creates a restoring moment or a disturbing moment. If the resultant side force acts behind G, the moment tends to reduce the yaw angle. That is positive static margin. If it acts ahead of G, the moment tends to increase the yaw angle. That is the unstable direction.

This example explains why rear axle contribution matters so much to directional stability. Dixon states the physical requirement: the rear axle restoring moment must exceed the front axle disturbing moment. The rear does not need to be mysterious or magical. It is a side force at a lever arm behind G. If its stiffness and lever arm are large enough relative to the front contribution, the resultant force acts behind G and the car tends to come back from the small yaw disturbance.

Keep this example separate from the skidpad force split. Static margin is about the response to a yaw angle with no steering and no yaw rate. The skidpad balance is about steady centripetal force with zero net yaw moment. They use the same pieces, but they are not the same problem. If you mix them together, you will misread both the equations and the car.

Worked example: yaw rate as damping, not just rotation

A third component case adds yaw rate. With the car rotating about G, the front axle and rear axle do not have the same local velocity direction. The axle ahead of G sees a velocity component from yaw rate proportional to its distance a. The axle behind G sees a corresponding contribution proportional to b. Dixon gives the small-angle slip-angle effects as lever-arm times yaw rate over speed.

Those yaw-rate slip-angle increments create tire side forces. Because they act at the front and rear lever arms, they create a yaw moment. Dixon identifies this moment as yaw damping because it opposes the yawing velocity. The important driver-analysis point is that yaw rate is not merely the car rotating. In the linear bicycle model, yaw rate creates its own tire slip-angle contributions, and those contributions feed back as a resisting yaw moment.

That also explains why speed appears in the denominator of the yaw damping contribution. For the same yaw rate and geometry, increasing forward speed reduces the slip-angle increment associated with yaw rate. The model therefore predicts a yaw-damping term that is inversely proportional to speed. You do not need to turn that into a setup conclusion here. The lesson is that steady-state balance and yaw response are speed-context problems, not just steering-angle problems.

Sub-skills: what you must be able to do on paper

First, you must be able to draw the reduced vehicle correctly. Put the axles on the centerline, mark G, and label a, b, and L. If you put the forces at G, you have already erased the yaw moment balance.

Second, you must be able to state the two balances in words before writing equations. The front and rear side forces must add to the centripetal force requirement. The front and rear yaw moments about G must cancel in steady state. If you cannot say those two sentences, the algebra will not protect you.

Third, you must be able to separate force share from slip-angle share. Force share comes from the lateral force requirement and the moment arms. Slip-angle share comes from the force each axle must make divided by that axle's cornering stiffness. Handling balance lives in the relationship between those two shares.

Fourth, you must be able to name which simplified case you are analyzing. Steer-only, yaw-angle-only, yaw-rate-only, and complete steady-state cornering are different constructions. Dixon uses these cases to build the full model because each reveals a different physical coefficient.

Fifth, you must be able to say when the model is being stretched. Small angles, low-slip linear tire behavior, neglected aero, neglected rolling resistance, neglected suspension compliance, and simplified tractive-force treatment are not permanent truths. They are assumptions that make the first model teachable.

Common mistakes

Mistake one is treating the bicycle model as a picture of a real bicycle. Good looks like saying that the model compresses left and right tires on each axle into one equivalent centerline wheel, while the vehicle itself remains a car with car-like tire side forces and no banking assumption.

Mistake two is writing only the lateral force balance. Fyf plus Fyr equals the required centripetal force, but that does not guarantee steady cornering. Good looks like writing the yaw moment balance immediately after the side-force balance and checking that the axle moments about G cancel.

Mistake three is assuming the front and rear axle forces must be equal. Good looks like using the lever arms. A shorter lever arm needs more force to produce the same moment, and a longer lever arm needs less force.

Mistake four is confusing side force with slip angle. Good looks like keeping two layers in your head: the path demands axle forces, then axle cornering stiffness determines the slip angles needed to generate those forces.

Mistake five is using the linear model outside its teaching range. Good looks like naming the assumptions: small angles, low slip angles, steady state, simplified tires, and neglected suspension or combined-force effects unless they are deliberately added.

Mistake six is mixing static margin with steady cornering balance. Good looks like saying that static margin is a yaw-disturbance question about where the resultant tire force acts relative to G, while steady cornering balance is a force-and-moment equilibrium for a chosen path.

Drill: two-minute bicycle-model rebuild

At your next study session or track debrief, do this drill three times, with a blank page each time. Each repetition should take two minutes, followed by one minute of checking.

In the first thirty seconds, draw the top view. Mark G, the front axle, the rear axle, a, b, and L. In the next thirty seconds, draw Fyf and Fyr at the axles and write the lateral force balance. In the next thirty seconds, write the yaw moment balance about G and solve the front-to-rear force ratio in words. In the final thirty seconds, add front and rear cornering stiffness and explain which axle needs more slip angle for its required force.

The success criterion is precise: by the third repetition, you should be able to rebuild the diagram and explain the two balances without looking at notes. If you skip the moment balance, the repetition does not count. If you cannot explain why equal axle forces are not generally required, the repetition does not count. If you jump to understeer gradient before force and moment balance, slow down and restart.

Calibration cues

On paper, improvement feels like fewer memorized fragments and more reconstruction. You can start with the drawing and regenerate the balances. You can look at a term and say whether it is a force term, a lever-arm moment term, a stiffness term, or a speed-radius demand term.

In data, the lesson prepares you to ask better questions rather than make instant conclusions. More steering for the same radius and speed points you toward front slip-angle demand. A car that rotates too readily points you toward the rear restoring moment, the front disturbing moment, and yaw damping. A balance change with speed points you toward the speed dependence of lateral force demand and yaw-rate contributions. Those are model-based questions, not guesses.

In driver language, improvement sounds like replacing vague balance words with axle-force language. Instead of saying the car just pushes, you ask whether the front axle needs more slip angle to make its required force. Instead of saying the rear is loose as a complete diagnosis, you ask whether the rear axle is failing to provide the restoring side-force moment required for the current state.

Where this model stops

The simple bicycle model is a foundation, not a complete race-car simulator. Dixon notes that neglecting the suspension is a serious practical limitation, even though the complete vehicle cornering stiffnesses illuminate the simple vehicle. Gillespie also points out that many design factors influence cornering force under lateral acceleration, with suspension and steering systems among the primary sources.

The model also becomes incomplete when longitudinal forces matter. Dixon's rear-drive and front-drive force diagrams show that tractive force changes the direction of the total tire force required for steady tangential speed. Braking analysis similarly brings in rotating wheels, braking forces, side forces, and yaw acceleration about the center of gravity. Those effects belong in expanded models.

So use the bicycle model honestly. Use it to learn the axle force balance, the yaw moment balance, static margin, and yaw damping. Do not use it to pretend that suspension compliance, load transfer, aero, braking, throttle, and nonlinear tires have disappeared from the real car.

Author Review

No quiz questions are attached to this lesson.

Sources

#	Document	Chunk	Pages	Score	Collection
1	Tires Suspension and Handling Second Edition Dixon John C	86d534d1-ae84-2f7d-df95-48c6a61170b5	350	1	uio_books_raw_v1
2	Fundamentals of vehicle dynamics Gillespie T. D. Thomas D.	7af13620-4bcb-7466-42d5-4350b58d1172	134	1	uio_books_raw_v1
3	Tires Suspension and Handling Second Edition Dixon John C	afb96fbe-faf8-d206-6cb6-48f74779d6c3	366	1	uio_books_raw_v1
4	Tires Suspension and Handling Second Edition Dixon John C	aa8d3621-f2d3-8f4c-6cf5-7b1109681135	368	1	uio_books_raw_v1
5	Tires Suspension and Handling Second Edition Dixon John C	3afaca2e-c041-dce0-75c1-3484e6e781a6	370	1	uio_books_raw_v1
6	Tires Suspension and Handling Second Edition Dixon John C	c3a60f4e-6336-28e0-8d3c-409f26426f3a	357	1	uio_books_raw_v1
7	Fundamentals of vehicle dynamics Gillespie T. D. Thomas D.	88cbdbfe-237b-b2ed-0f8b-09605b9839e3	139	1	uio_books_raw_v1
8	Brake Design and Safety Rudolf Limpert	96235c78-b994-d95c-ba54-eed29859d7a3	267	1	uio_books_raw_v1