Directional stability
Encyclopedia
Directional stability is stability of a moving body or vehicle about an axis which is perpendicular to its direction of motion. Stability of a vehicle concerns itself with the tendency of a vehicle to return to its original direction in relation to the oncoming medium (water, air, road surface, etc.) when disturbed (rotated) away from that original direction. If a vehicle is directionally stable, a restoring moment
is produced which is in a direction opposite to the rotational disturbance. This "pushes" the vehicle (in rotation) so as to return it to the original orientation, thus tending to keep the vehicle oriented in the original direction.
Directional stability is frequently called "weather vaning" because a directionally stable vehicle free to rotate about its center of mass is similar to a weather vane
rotating about its (vertical) pivot.
With the exception of spacecraft, vehicles generally have a recognisable front and rear and are designed so that the front points more or less in the direction of motion. Without this stability, they may tumble end over end, spin or orient themselves at a high angle of attack
, even broadside on to the direction of motion. At high angles of attack, drag
force
s may become excessive, the vehicle may be impossible to control, or may even experience structural failure. In general, land, sea, air and underwater vehicles are designed to have a natural tendency to point in the direction of motion.
.
The first stage of studying the stability of a road vehicle is the derivation of a reasonable approximation to the equations of motion.
The diagram illustrates a four wheel vehicle, in which the front axle is located a metres ahead of the centre of gravity and the rear axle is b metres aft of the cg. The body of the car is pointing in a direction
(theta) whilst it is travelling in a direction 
(psi). In general, these are not the same. The tyre treads at the region of contact point in the direction of travel, but the hubs are aligned with the vehicle body, with the steering
held central. The tyres distort as they rotate to accommodate this mis-alignment, and generate side forces as a consequence.
The net side force Y on the vehicle is the centripetal force causing the vehicle to change the direction it is traveling:
where M is the vehicle mass
and V the speed.
The angles are all assumed small, so the lateral force
equation is:
The rotation of the body subjected to a yawing moment N is governed by:
where I is the moment of inertia
in yaw.
The forces and moments of interest arise from the distortion of the tyres. The angle between the direction the tread is rolling and the hub is called the slip angle
. This is a bit of a misnomer, because the tyre as a whole does not actually slip, part of the region in contact with the road adheres, and part of the region slips. We assume that the tyre force is directly proportional to the slip angle (phi). This is made up of the slip of the vehicle as a whole modified by the angular velocity of the body. For the front axle:
whilst for the rear axle:
Let the constant of proportionality be k. The sideforce is, therefore:
The moment is:
Denoting the angular velocity
, the equations of motion are:
Let
(beta), the slip angle for the vehicle as a whole:
Eliminating
yields the following equation in 
:
This is called a second order linear homogenous equation, and its properties form the basis of much of control theory
.
The coefficient of
will be called the 'damping
' by analogy with a mass-spring-damper which has a similar equation of motion.
By the same analogy, the coefficient of
will be called the 'stiffness', as its function is to return the system to zero deflection, in the same manner as a spring.
The form of the solution depends only on the signs of the damping and stiffness terms. The four possible solution types are presented in the figure.
The only satisfactory solution requires both stiffness and damping to be positive.
The damping term is:
The tyre slip coefficient k is positive, as are the mass, moment of inertia and speed, so the damping is positive, and the directional motion should be dynamically stable.
The stiffness term is:
If the centre of gravity is ahead of the centre of the wheelbase
(
, this will always be positive, and the vehicle will be stable at all speeds. However, if it lies further aft, the term has the potential of becoming negative above a speed given by:
Above this speed, the vehicle will be directionally unstable.
If the rear tyres produce no significant forces, the side force and yawing moment become:
The equation of motion becomes:
The coefficient of
is negative, so the vehicle will be unstable.
Consider the effect of faulty tyres at the front. The Side force and yawing moment become:
The equation of motion becomes:
The coefficient of
is positive, so the vehicle will be stable but unsteerable.
It follows that the condition of the rear tyres is more critical to directional stability than the state of the front tyres. Also, locking the rear wheels by applying the handbrake, renders the vehicle directionally unstable, causing it to spin. Since the vehicle is not under control during the spin, the 'handbrake turn' is usually illegal on public roads.
The side force becomes:
where
(eta) is the steering deflection. Similarly, the yawing moment becomes:
Including the steering term introduces a forced response:
The steady state response is with all time derivatives set to zero. Stability requires that the coefficient of
must be positive, so the sign of the response is determined by the coefficient of 
:
This is a function of speed. When the speed is low, the slip is negative and the body points out of the corner (it understeer
s). At a speed given by:
The body points in the direction of motion. Above this speed, the body points into the corner (oversteers).
As an example:
Evidently moving the centre of gravity forwards increases this speed, giving the vehicle a tendency to understeer
.
Note: Installing a heavy, powerful engine in a light weight production vehicle designed around a small engine increases both its directional stability, and its tendency to understeer. The result is an overpowered vehicle with poor cornering performance.
Even worse is the installation of an oversized power unit into a rear engined production vehicle without corresponding modification of suspension or mass distribution, as the result will be directionally unstable at high speed.
The amateur mechanic's dream car can readily become an uncontrollable nightmare.
A full analysis should also take account of the suspension
response.
The complete analysis is essential for the design of high performance road vehicles, but is beyond the scope of this article.
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
is produced which is in a direction opposite to the rotational disturbance. This "pushes" the vehicle (in rotation) so as to return it to the original orientation, thus tending to keep the vehicle oriented in the original direction.
Directional stability is frequently called "weather vaning" because a directionally stable vehicle free to rotate about its center of mass is similar to a weather vane
Weather vane
A weather vane is an instrument for showing the direction of the wind. They are typically used as an architectural ornament to the highest point of a building....
rotating about its (vertical) pivot.
With the exception of spacecraft, vehicles generally have a recognisable front and rear and are designed so that the front points more or less in the direction of motion. Without this stability, they may tumble end over end, spin or orient themselves at a high angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
, even broadside on to the direction of motion. At high angles of attack, drag
Drag coefficient
In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or...
force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s may become excessive, the vehicle may be impossible to control, or may even experience structural failure. In general, land, sea, air and underwater vehicles are designed to have a natural tendency to point in the direction of motion.
Example - Road Vehicle
Arrows, darts, rockets and airships have tail surfaces to achieve stability. A road vehicle does not have elements specifically designed to maintain stability, but relies primarily on the distribution of massMass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
.
Introduction
These points are best illustrated with an example which is familiar to most readers - the humble motor car.The first stage of studying the stability of a road vehicle is the derivation of a reasonable approximation to the equations of motion.
The diagram illustrates a four wheel vehicle, in which the front axle is located a metres ahead of the centre of gravity and the rear axle is b metres aft of the cg. The body of the car is pointing in a direction
(theta) whilst it is travelling in a direction (psi). In general, these are not the same. The tyre treads at the region of contact point in the direction of travel, but the hubs are aligned with the vehicle body, with the steeringSteering
Steering is the term applied to the collection of components, linkages, etc. which will allow a vessel or vehicle to follow the desired course...
held central. The tyres distort as they rotate to accommodate this mis-alignment, and generate side forces as a consequence.
The net side force Y on the vehicle is the centripetal force causing the vehicle to change the direction it is traveling:
where M is the vehicle mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
and V the speed.
The angles are all assumed small, so the lateral force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
equation is:
The rotation of the body subjected to a yawing moment N is governed by:
where I is the moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
in yaw.
The forces and moments of interest arise from the distortion of the tyres. The angle between the direction the tread is rolling and the hub is called the slip angle
Slip angle
In vehicle dynamics, slip angle or sideslip angle is the angle between a rolling wheel's actual direction of travel and the direction towards which it is pointing...
. This is a bit of a misnomer, because the tyre as a whole does not actually slip, part of the region in contact with the road adheres, and part of the region slips. We assume that the tyre force is directly proportional to the slip angle (phi). This is made up of the slip of the vehicle as a whole modified by the angular velocity of the body. For the front axle:
whilst for the rear axle:
Let the constant of proportionality be k. The sideforce is, therefore:
The moment is:
Denoting the angular velocity
, the equations of motion are:Let
(beta), the slip angle for the vehicle as a whole:Eliminating
yields the following equation in :This is called a second order linear homogenous equation, and its properties form the basis of much of control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
.
Stability Analysis
We do not need to solve the equation of motion explicitly to decide whether the solution diverges indefinitely or converges to zero following an initial perturbation. The form of the solution depends on the signs of the coefficients.The coefficient of
will be called the 'dampingDamping
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
' by analogy with a mass-spring-damper which has a similar equation of motion.
By the same analogy, the coefficient of
will be called the 'stiffness', as its function is to return the system to zero deflection, in the same manner as a spring.The form of the solution depends only on the signs of the damping and stiffness terms. The four possible solution types are presented in the figure.
The only satisfactory solution requires both stiffness and damping to be positive.
The damping term is:
The tyre slip coefficient k is positive, as are the mass, moment of inertia and speed, so the damping is positive, and the directional motion should be dynamically stable.
The stiffness term is:
If the centre of gravity is ahead of the centre of the wheelbase
Wheelbase
In both road and rail vehicles, the wheelbase is the distance between the centers of the front and rear wheels.- Road :In automobiles, the wheelbase is the horizontal distance between the center of the front wheel and the center of the rear wheel...
(
, this will always be positive, and the vehicle will be stable at all speeds. However, if it lies further aft, the term has the potential of becoming negative above a speed given by:
Above this speed, the vehicle will be directionally unstable.
Relative Effect of Front and Rear Tyres
If for some reason (incorrect inflation pressure, worn tread) the tyres on one axle are incapable of generating significant lateral force, the stability will obviously be affected. Assume to begin with that the rear tyres are faulty, what is the effect on stability?If the rear tyres produce no significant forces, the side force and yawing moment become:
The equation of motion becomes:
The coefficient of
is negative, so the vehicle will be unstable.Consider the effect of faulty tyres at the front. The Side force and yawing moment become:
The equation of motion becomes:
The coefficient of
is positive, so the vehicle will be stable but unsteerable.It follows that the condition of the rear tyres is more critical to directional stability than the state of the front tyres. Also, locking the rear wheels by applying the handbrake, renders the vehicle directionally unstable, causing it to spin. Since the vehicle is not under control during the spin, the 'handbrake turn' is usually illegal on public roads.
Steering Forces
Deflecting the steering changes the slip angle of the front tyres, generating a sideforce. With conventional steering, the tyres are deflected by different amounts, but for the purposes of this analysis, the additional slip will be considered equal for both front tyres.The side force becomes:
where
(eta) is the steering deflection. Similarly, the yawing moment becomes:Including the steering term introduces a forced response:
The steady state response is with all time derivatives set to zero. Stability requires that the coefficient of
must be positive, so the sign of the response is determined by the coefficient of :
This is a function of speed. When the speed is low, the slip is negative and the body points out of the corner (it understeer
Understeer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Simply put, oversteer is what occurs when a car turns by more than the amount commanded by the driver...
s). At a speed given by:
The body points in the direction of motion. Above this speed, the body points into the corner (oversteers).
As an example:
-
- with k=10kN/radian, M=1000kg, b=1.0m, a=1.0m, the vehicle understeers below 11.3mph.
Evidently moving the centre of gravity forwards increases this speed, giving the vehicle a tendency to understeer
Understeer
Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Simply put, oversteer is what occurs when a car turns by more than the amount commanded by the driver...
.
Note: Installing a heavy, powerful engine in a light weight production vehicle designed around a small engine increases both its directional stability, and its tendency to understeer. The result is an overpowered vehicle with poor cornering performance.
Even worse is the installation of an oversized power unit into a rear engined production vehicle without corresponding modification of suspension or mass distribution, as the result will be directionally unstable at high speed.
The amateur mechanic's dream car can readily become an uncontrollable nightmare.
Limitations of the Analysis
The forces arising from slip depend on the loading on the tyre as well as the slip angle, this effect has been ignored, but could be taken into account by assuming different values of k for the front and rear axles. Roll motion due to cornering will redistribute the tyre loads between the nearside and offside of the vehicle, again modifying the tyre forces. Engine torque likewise re-distributes the load between front and rear tyres.A full analysis should also take account of the suspension
Suspension (vehicle)
Suspension is the term given to the system of springs, shock absorbers and linkages that connects a vehicle to its wheels. Suspension systems serve a dual purpose — contributing to the car's roadholding/handling and braking for good active safety and driving pleasure, and keeping vehicle occupants...
response.
The complete analysis is essential for the design of high performance road vehicles, but is beyond the scope of this article.
See also
- Relaxed stabilityRelaxed stabilityIn aviation, relaxed stability is the tendency of an aircraft to change its attitude and angle of bank of its own accord. An aircraft with relaxed stability will oscillate in simple harmonic motion around a particular attitude at an increasing amplitude....
- Car handlingCar handlingAutomobile handling and vehicle handling are descriptions of the way wheeled vehicles perform transverse to their direction of motion, particularly during cornering and swerving. It also includes their stability when moving at rest. Handling and braking are the major components of a vehicle's...
- Flight dynamicsFlight dynamicsFlight dynamics is the science of air vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of mass, known as pitch, roll and yaw .Aerospace engineers develop control systems for...
- Longitudinal static stabilityLongitudinal static stabilityLongitudinal static stability is the stability of an aircraft in the longitudinal, or pitching, plane during static conditions. This characteristic is important in determining whether an aircraft will be able to fly as intended...
- Hunting oscillationHunting oscillationHunting oscillation is an oscillation, usually unwanted, about an equilibrium. The expression came into use in the 19th century and describes how a systems 'hunts' for equilibrium...