Single track

The linear single-track model is the simplest model concept to explain the steady state and transient lateral dynamics of double-track motor vehicles. The track model was developed by Riekert and Schunck in 1940 and used for the analysis of the steering and disturbance behavior in crosswinds. To date, the single-track model is the essential theory basis for vehicle engineers with a specialty in driving dynamics. The most widely used is the linear single-track model found in the ESP control devices, where it is used to driver input detection.

Since still largely behave car on dry pavement up to a lateral acceleration of approximately 4 m/s2 linear, the transverse dynamic behavior can be explained in this area by the linear single-track model approximation.

In addition to the linear single-track model, there are Einspurmodelle with different levels of detail eg nonlinear Einspurmodelle or Einspurmodelle with additional Wankfreiheitsgrad.

Below are details on the linear track model with two degrees of freedom ( yaw rate, sideslip angle ). It is treated without any external disturbances of the special case of the steered front axle of the vehicle.

  • 6.1 natural frequency and damping
  • 6.2 transmission behavior
  • 6.3 Frequency Response
  • 7.1 Equations of Motion
  • 7.2 Stationary vehicle reactions
  • 7.3 Calculation of the transfer functions

Areas of application

  • Plausibility from experimental and simulation results.
  • Driver request detection in vehicle dynamics control systems.
  • Identification of parameters from measured data (eg, natural frequency, damping Lehr ).
  • Separation of driving and noise components (eg crosswind ) from measured data.
  • Stability considerations in the closed control loop.
  • Development of vehicle dynamics control systems eg rear axle.

Assumptions

  • Both wheels of an axle are combined to form a wheel.
  • Kinematics and elasto- axis are packed into the tire.
  • Linear tire behavior.
  • Self-aligning moments are neglected.
  • No tire run-in behavior.
  • No waver.
  • The change in speed can be treated quasi-stationary.
  • Small angle cos ( φ ) ≈ 1, sin ( φ ) ≈ φ

Kinematic relations

The planar motion of a rigid body can always be considered as rotation about the instantaneous center M. Therefore applies to the speed of gravity

The yaw rate respectively.

The velocity vectors of the wheel perpendicular to the polar rays. The included angle of the polar rays is therefore the difference of the angular velocity vector of the front axle and rear axle.

Under the condition of small angle of the included angle, the ratio of wheel base and pole distance R. In the case of a steered only at the front axle of the vehicle Vorderachseinschlag DELTA.V is equal to the steering angle δ, so that the relationship

Results. The steering angle is composed of the Ackermann angle and the skew angle difference between front and rear axle.

The skew angle can be derived from the degrees of freedom ( state variables ) to calculate yaw rate and side slip angle and the steering angle:

Forces and moments

Neglecting wind forces and moments, so act as external forces the Achsseitenkräfte on the vehicle a. These are proportional to the oblique running angle of the linear single-track model:

For given forces, the equations of motion are ( linear momentum and angular momentum ):

With the centripetal or radial acceleration is known, points perpendicular to the velocity vector in focus towards instantaneous.

In terms of the degrees of freedom of the single-track model applies to the radial acceleration

Stationary behavior

With a stationary drive, the moment of external forces relative to the center of gravity to zero. It is therefore

Combining the last equation with the pulse rate, the lateral forces as a function of the radial acceleration may be expressed as:

Inserted into the equation obtained from the kinematic conditions for the steering angle results

The expression in brackets is referred to as self-steering gradient (EG). Instead of the compact notation for the EC (top), and an equation can be used in only enter into direct parameters:

The stationary steering angle is made up of a share depends only on wheelbase and radius ( Ackermann angle) and a share proportional to the radial acceleration together. The self-steering behavior is distinguished according to the sign of the EC:

All of today's cars are designed to understeer. About Controlling vehicles can become unstable.

Yaw gain

The yaw gain is the ratio between steady yaw rate and steering wheel angle.

With (this is the overall steering ratio ) applies to the stationary steering wheel angle

In steady-state circular driving, applies since swimming angular velocity becomes zero. The radius can also be expressed in. Used in the above equation is obtained for the yaw gain

The picture shows possible paths depending on the sign of the self-steering gradient. Under Controlling vehicles have a maximum of yaw gain, which at the characteristic speed

Occurs. In the case of EC <0, the denominator of the yaw gain can be zero. The speed at which this occurs is called the critical speed

Referred to. The yaw gain grows at this point beyond all limits, the vehicle is unstable.

The maximum yaw gain in relation to the steering wheel angle has to be controlled vehicles under the value

Most cars have maximum yaw gains in the range of 0.2 1 / s and 0.4 1 / s As a characteristic of the maximum yaw gain describes the Agilitätseindruck a vehicle at moderate speeds ( highway ). The characteristic velocity is in most cars in the range of about 70 km / h and 110 km / h

The initial slope of the yaw gain on the driving speed is referred to as static steering sensitivity. It is a measure for the maneuverability of a vehicle at low speeds. It has the value

And depends only on overall steering ratio and wheelbase. Short wheelbase vehicles are therefore perceived as more maneuverable at low speeds.

Sideslip angle

The sideslip angle can already be derived from the kinematic relations. The following applies:

The term is referred to as Schwimmwinkelgradient ( SG). Small Schwimmwinkelgradienten are the prerequisite for safe, stable handling. Main influencing factor is the choice of tires on the rear axle.

Analogous to the yaw gain can also calculate a floating angle reinforcement. After some transformations we have:

At high slip angle reinforcements are moving dynamically undesirable because they contribute to an unsafe driving behavior at high speeds.

Dynamic behavior

In the dynamic behavior of interest on or decay processes, as well as the response to certain test signals. The most important are the steering input and the sinusoidal excitation. These processes are characterized by their transfer functions.

For the calculation of these properties it is recommended to change to the state space representation:

Natural frequency and damping

One or decaying are determined by natural frequency and damping. These can be calculated with the help of eigenvalues. The eigenvalues ​​resulting from the characteristic polynomial

Written out:

The solution of the characteristic polynomial is

The stability criterion according to Hurwitz states that all coefficients of the polynomial must be positive. Only the constant term can be negative. It follows:

In the vehicle parameters expressed:

The stability condition of the track model is thus:

This condition was derived already at the critical speed. The stationary characteristic of self-steering gradient is thus also an important measure of stability.

Usual car at low speeds have real eigenvalues ​​, at medium to high speeds complex conjugate eigenvalues. In terms of driving dynamics relevant driving speeds can be assumed by the complex conjugate case. The eigenvalues ​​can be interpreted as follows:

Here are

By comparison with the eigenvalues ​​yields:

Given the following vehicle data:

Thus the eigenvalues ​​shown in the image depending on the speed yield. From the eigenvalues ​​of natural frequencies and damping can be calculated. Desirable high natural frequencies and high attenuation, but under the boundary conditions of the vehicle not vote can be implemented consistent.

Transmission behavior

All quantities of interest ( output variables ) can be calculated by means of the state variables and the input of steering wheel angle. To the Laplace transform is used ( complex variable). The same symbols are used for the sake of simplification in the image and in the time domain.

As an example, the lateral acceleration may be mentioned, which equation has been shown in forces and moments. The matrices and can then be written:

With:

Here is the vector of transfer functions of slip angle and yaw rate.

It can be shown that each transfer function in the form:

Can be accommodated.

The denominator is the same for all transfer functions and has the form:

One can calculate analytically the transfer functions with the help of the vehicle parameters. Thus, for example, bring the transfer function of the yaw rate to the following form:

The amplification factor is the already derived yaw gain. The meter time constant is calculated as

And thus corresponds to the product of speed and Schwimmwinkelgradient.

Frequency response

Substituting the complex variable is obtained from the transfer functions of the frequency response. The frequency response is a complex function of. From the amount we obtain the amplitude response, from the angle of the phase transition. The frequency responses are determined most easily in the computer, but they can also be specified as a function of the vehicle parameters.

Most often, the frequency response of yaw rate and lateral acceleration can be calculated or measured during driving. In the measurement of the vehicle is taken into account that the acceleration is a function of location, and thus must be converted from the measuring point to the center of gravity.

With the vehicle data from the table above, the frequency response of yaw rate and lateral acceleration show a strong dependence on the driving speed.

The strong increase in the amplitude response of the yaw rate is in addition to the decreasing attenuation on the driving speed can also be explained with the meter time constants. An effective measure for improving the dynamic properties is as small as possible Schwimmwinkelgradient.

Mathematical derivation

Equations of motion

The motion equations are expressed in terms of the state variables:

The already introduced above matrices are therefore:

In state space representation:

Stationary vehicle reactions

In case of inpatient ride the derivatives of the state variables to zero. The state variables thus calculated to be

Expressed in terms of the components of the matrix, and:

After a few transformations (see also # Regular matrix formula for 2x2 matrices ) yields:

Calculation of the transfer functions

Starting from the equation

Results, expressed in the components of the matrices and,

With

With the previously calculated gains are obtained:

Eigenfrequency Step response
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