Vehicle Dynamics Modeling

Suspension design plays a critical role in ride and handling of a vehicle, with the ability to tune the response of the system to address different road and steering inputs. Below are a few different models that are commonly used to understand the response modes associated with passenger vehicles. Click on the tabs to get a breakdown of each model.

Vehicle ride

Vehicle-road interactions greatly affect the experience of drivers and passengers. Developing vehicle ride models allows for informed decision making with regards to suspension parameters. Included in this section are both the famous Quarter Car mode, and the Pitch-Plane model, which describe how suspension systems respond to vertical road inputs and the resulting vertical and pitching motion.

The quarter car model is an idealized two-mass system that allows for an approximated dynamic response of a vehicle suspension to heave based inputs. In the system, the sprung mass (chassis/body) is connected to the unsprung mass (wheel hub assembly) by a spring and damper. The hub is then “connected” to the ground by the tire which can be represented by another spring and damper.

In this assembly, the downwards direction is the +z direction, where , is the displacement of the sprung mass, is the displacement of the unsprung mass, and is the displacement of the ground. The spring stiffnesses, and , are the suspension and tire stiffnesses respectively, and and are the damping coefficients for the suspension and tire. The masses, and , are the sprung and unsprung masses.

For this model, the following values are used:

** This parameter is to be varied.

Dynamics

The quarter car model can be described in both the time domain (state-space) or the Laplace s-domain. While the s-domain can be useful for analyzing the system and allow for the use of Simulink, it is more difficult to use in cases where non-linear damping is used. Therefore, both versions of the model are developed below.

State-Space

The forces within the system are described by the equations below.

The system can then be rearranged and presented in a state-space form.

Laplace s-domain

Using Laplace transform, Equation 1 and 2 can be rewritten to be expressed in terms of s.

This can be expressed in matrix form:

Natural Frequencies

The natural frequencies of the quarter car model can be analyzed to determine how changes in stiffness and damping affect the driver comfort. Human sensitivity to vibrations has been shown to be the highest in the frequency band of 4-8 Hz. Therefore, it is important to analyze the suspension damping and stiffness parameters to minimize frequencies in this range while maintaining low amplitudes for natural frequencies. Using the state-space model, the parameters in the table above and Cs = 5 lbs*sec/in, the following plot was generated using a Bode plot function in Matlab to determine the amplitude ratio associated with a given frequency.

In the left plot, the first large peak is the natural frequency associated with the sprung mass, while the second large peak is the natural frequency for the unsprung mass (wheel hop). In the right plot, increasing the suspension damping coefficient decreases the amplitude ratio for the sprung mass natural frequency, however this comes at the tradeoff of increased amplitude for frequencies within the sensitivity band. The goal is to minimize natural frequency amplitude ratio and minimize the amplitude ratios of frequencies within the sensitivity band.

To calculate the performance effects of increasing the damping coefficient, amplitude ratios of the damped configuration were calculated as functions of damping coefficient, and then compared to amplitude ratios of the undamped configuration.

By tracking the percent change in the amplitude ratios, the return on increased damping can be quantified.

(Left plot)

For the amplitude ratio of the sprung mass natural frequency, the percent improvement is greater at small values of Cs, but returns begin to quickly decline as Cs gets large. The rate of increase in the maximum amplitude ratios within the sensitivity band is shown to be smaller at Cs = 4 lbs-sec/in, but has a steeper linear percent change for greater Cs values.

(Right plot)

To choose an optimal Cs for this analysis, the percent changes in amplitude ratios were compared. At the chosen value, Cs = 4 lbs-sec/in, there is a 43% increase in the maximum amplitude ratio experienced in the sensitivity band, but a 53% decrease in the amplitude ratio for the sprung mass natural frequency. This point marks the start of reduced performance returns based on the above plots.

In the future, stronger optimization and weighting functions should be used to make more informed decisions, however, this general approach provides a starting point for the analysis.

Simulating Responses to Road Inputs

There are six different input cases for that will be considered:

A 1-inch step function at 0.5 Hz
A ¼ inch step function at 0.5 Hz
A +/- 1-inch sine wave input at 1 Hz
A +/- ¼ inch sine wave at 10 Hz
A 2-inch pothole 3 ft wide at 60 mph
A 2-inch pothole 3 ft wide at 30 mph

These input cases are shown in the figures below.

Using the linear damping model and the Cs value (4 lbs-sec/in) found in the previous section, the system was tested using the input functions to better understand the response to different types of road inputs.

A couple key observations include high overshoot for step functions with a longer period, high amplitude ratios for the 1 Hz wave, and undershoot for the step functions with short periods that are less than the system rise time. The high overshoot and longer settling times suggests that further tuning should be conducted to find a more appropriate damping parameter. In addition, the introduction of non-linear damping and rebound rate control are potential ways to optimize system response.

Vehicle Handling

Models coming soon!