Comparing critically damped, underdamped and overdamped steering responses in ADAMS and Excel

Portfolio Section 4 Steering Response of a System
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Joseph Douglas Pearce 15056080
SECTION 4 - STEERING RESPONSE OF A SYSTEM
A comparison between Critically Damped, Underdamped and Overdamped steering responses.
OVERVIEW
For this report, we will be comparing two methods of finding the response characteristics of a vehicle. In
the first section, we will be creating an Excel spreadsheet that allows us to calculate the yaw-rate of the
vehicle with respect to time. Secondly, we will be creating a ‘bicycle’ model using the ADAMS software
program that should allow us to achieve the same thing. Finally we will be comparing the results.
PHYSICAL CHARACTERISTICS
For this analysis we will be using the values given for the generic small hatchback as used in Race Car
Vehicle Dynamics (Milliken & Milliken, 1995). These values are presented as follows:
PHYSICAL CONSTANTS
Name Shorthand Units Values
Mass m kg 906
Polar Moment Inertia Iz kg*m^2 1556
Weight Distribution F/R - 65/35
CoG Front a m 1.22
Wheelbase l m 2.54
Cornering Stiffness Front Cf N/° -1068
Cornering Stiffness Rear Cr N/° -890
All of these constants, with the exception of the F/R and a values shall remain constant throughout this
analysis. It is necessary to alter these values in order to affect the handling characteristics of the vehicle.
We must also consider two user input parameters. The reader is invited to alter them, but due to the
limited scope of this report, they shall remain constant throughout. In order to properly model the
response we need to know the velocity of the vehicle and the steering input. The velocity was chosen to
be sufficiently fast as to provoke an interesting response – 120kph (33.3m/s – 74.4mph) In both models
the steering input was chosen to be a step input of 5°.
SCOPE
We will be investigating three handling characteristics of the small hatchback – Critically Damped
response, Underdamped response and Overdamped response. As we have previously stated, to achieve
this we will be moving the centre of gravity (CoG) forwards and backwards across the car to provoke the
desired response. The values chosen and their respective Damping Ratio (ζ) values are tabulated below:

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Response Wheelbase (l) CoG Front (a) CoG Rear (b) F/R
Damping Ratio
(ζ)
Critical
2.54m
1.15m 1.39m 55/45% 0.99
Underdamped 0.89m 1.65m 65/35% 0.65
Overdamped 1.27m 1.27m 50/50% 1.69
EXCEL CONSTRUCTION
Our three spreadsheets are labelled ‘Critically Damped’, ‘Underdamped’ and ‘Overdamped’. In each we
have the following array of cells:
PHYSICAL CONSTANTS:
Identical to the table that appears in the ‘Physical Characteristics’ section of this report. This
covers all the fixed parameters of the vehicle.
DEPENDENT CONSTANTS:
Basic manipulation of the data in the ‘Physical Constants’ section. This allows us to find the
length b and the Cf and Cr values in N/rads.
DERIVATIONS:
Physical data representing the motion of the vehicle. Yβ, Yr and Yδ refer to the lateral
performance of the car with the subscripts β, δ and r acting with respect to degree of slip, yaw
velocity and steering input respectively. The Nβ, Nr and Nδ values provide the same information
but for yaw acceleration instead of lateral acceleration.
𝑌𝛽 = 𝐶𝑓 + 𝐶𝑟; 𝑌𝑟 =
1
𝑉
(𝐶 𝑓 ∗ 𝑎 − 𝐶𝑟 ∗ 𝑏); 𝑌𝛿 = −𝐶𝑓
𝑁𝛽 = (𝐶 𝑓 ∗ 𝑎 − 𝐶𝑟 ∗ 𝑏); 𝑁𝑟 =
1
𝑉
(𝐶 𝑓 ∗ 𝑎2
− 𝐶𝑟 ∗ 𝑏2
); 𝑁𝛿 = −𝐶𝑓 ∗ 𝑎
(Balkwill, 2015)
INPUT PARAMETERS:
The required inputs for Velocity (V) in m/s and Steering Angle (δ) in °s and rads.
DERIVED PARAMETERS:
The calculated parameters necessary to find the response of the vehicle. As the Critically
Damped, Under- and Overdamped responses require different factors, this varies between
sheets:

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Coefficient Description Equation
𝐼 Inertia of System 𝐼 𝑍
𝑐 Damping of System 𝑁𝑟 +
(𝐼 ∗ 𝑌𝛽)
𝑚 ∗ 𝑉
⁄
𝑘 Stiffness of System 𝑁𝛽 + (
𝑌𝛽 ∗ 𝑁𝑟
𝑚 ∗ 𝑉
⁄ −
𝑁𝛽 ∗ 𝑌𝑟
𝑚 ∗ 𝑉
⁄ )
𝐶1 Coefficient of Steering Increase 𝑁𝛿
𝐶2 Coefficient of Steering Angle
𝑌𝛿 ∗ 𝑁𝛽
𝑚 ∗ 𝑉
⁄ −
𝑁𝛿 ∗ 𝑌𝛽
𝑚 ∗ 𝑉
⁄
𝑟̇0 Initial Yaw Acceleration
𝐶𝑓 ∗ 𝛿 ∗ 𝑎
𝑙
⁄
𝑟∞ Steady State Yaw Rate 𝐶2 ∗ 𝛿
𝑘
⁄
𝜔 𝑛 Natural Frequency √ 𝑘
𝐼⁄
𝑐 𝑐 Critical Damping 2 ∗ 𝐼 ∗ 𝜔 𝑛
𝜁 Damping Ratio 𝑐
𝑐 𝑐⁄
𝜔 𝑑 Damped Natural Frequency 𝜔 𝑛 ∗ √1 − 𝜁2
𝛷 Phase Angle tan−1
(
𝜔 𝑑
𝜔 𝑛 ∗ 𝜁⁄ )
𝑋 Yaw Constant (Critical, Under)
𝐶2 ∗ 𝛿
𝑘 ∗ sin(𝛷)⁄
𝑓 Response Coefficient (Over) (𝜁 − √𝜁2 − 1) ∗ 𝜔 𝑛
𝑔 Response Coefficient (Over) (𝜁 + √𝜁2 − 1) ∗ 𝜔 𝑛
ℎ Overdamped Response Ratio
𝑓
𝑔⁄
𝐴 Yaw Constant (Over)
𝑟̇0 + 𝑟∞ ∗ 𝑓
𝑔 − 𝑓⁄
𝐵 Yaw Constant (Over) 𝑟∞ + 𝐴
RESULTS:
The results are presented over the 5 seconds after the steering input was applied, calculated
with a δt of 0.01 seconds (100Hz). Displayed are the two components and total Yaw Rate (r).
GRAPH:
Each sheet contains a graph displaying the response of the Yaw Rate against Time.

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ADAMS CONSTRUCTION
We begin by creating our three blocks. The physical characteristics for these are implemented using the
standard build tools in ADAMS. It should be noted that the mass and inertial values for the ‘tyres’ have
been set to zero. The body must be constrained so as to translate in two dimensions and rotate around
its vertical axis. This is done by placing a planar joint at the body CoG. Then we can build the two ‘tyres’.
These are affixed to the body using two different joints – a fixed (padlock) between the rear and the
body, and a revolute (hinge) between the front and the body. This allows the front tyre to rotate in
order to steer the vehicle:
Figure 1 – Construction of body with all three parts and joints visible
We can see why the model is colloquially referred to as a ‘bicycle’. In this case, we have decided that +ve
y is forward motion.
MOTIONS AND FORCES
The front tyre is to be rotated using a step-steer input starting after 2 seconds, lasting for 0.01 seconds
and rotating from 0 to 5 degrees. The following expression was used to define the steering input:
Function 1 – Steering input
We wish to maintain a steady speed throughout the simulation. Our initial velocity is set at 33m/s but
we need to add a speed controller to maintain this velocity as the vehicle is turning. We add an
accelerative force (ACCELERATION_F) equal to 60000N and a retarding force (RETARDING_F) equal to
53*Velocity2
. This gives us the following model speed:

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Figure 2 – Velocity/Time plot for the ADAMS bicycle model
As we can see, when the steering input is applied the velocity drops from ≈ 33.65 to ≈ 33.05 m/s.
CENTRE OF GRAVITY POSITION
In order to change the position of the CoG, it is necessary to manually move the related marker
BODY.CM around in ADAMS. In order to speed up the process, three separate files have been created:
BICYCLE_CRITICAL, BICYCLE_UNDER and BICYCLE_OVER, each with the CoG moved to the pre-
determined position.
SIMULATION
The step-steering input is not applied until after 2 seconds. As our Excel calculations have been
conducted over 5 seconds from steering input, we will run the simulation for a total of 7 seconds. As
with the Excel spreadsheet, calculations were made at 100Hz, resulting in 700 calculations in total.
POST PROCESSING
The raw data was exported into the Excel file using the Export function in ADAMS post-processor. This
was to enable us to directly compare the data in the same axes and using the same plotting tools.

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RESULTS
Figure 3 – Plot showing Yaw Rate/Time for all ADAMS simulations
Figure 4 – Plot showing Yaw Rate/Time for all Excel simulations
0
10
20
30
40
50
60
70
80
0 200 400 600
YawRate(°/s)
Time (s)
ADAMS Models
Critically Damped
Underdamped
Overdamped
0
10
20
30
40
50
60
70
0 2 4 6
YawRate(°/s)
Time (s)
Excel Models
Critically Damped
Underdamped
Overdamped

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Figure 5 – Plot showing Yaw Rate/Time both Critically Damped simulations
Figure 6 - Plot showing Yaw Rate/Time both Underdamped simulations
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6
YawRate(°/s)
Time (s)
Critically Damped (ζ = 0.99)
Excel
ADAMS
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
YawRate(°/s)
Time (s)
Underdamped (ζ = 0.65)
Excel
ADAMS

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Figure 7 - Plot showing Yaw Rate/Time both Overdamped simulations
ANALYSIS
The first thing we can see is that the response of the three systems are what we might expect for the
Critical, Under- and Overdamped systems respectively; the Overdamped system has a similar form to
the Critically Damped, but takes longer to settle, with the Underdamped system overshooting before
settling to a lower steady Yaw Rate. What we can also see is that the absolute values of steady Yaw
Rate differ greatly between the same model using ADAMS and Excel:
 Critically Damped ≈ ± 13°/s
 Underdamped ≈ ± 5°/s
 Overdamped ≈ ± 10°/s
What this suggests is that ADAMS and our Excel program go about this solution in different ways. This in
itself is not surprising – ADAMS is a software that has been created specifically for this express purpose
and our Excel program is in its first iteration. The correlation between the responses from our graphs is
encouraging, but by no means complete. One interesting comparison is between the Overdamped
results from Excel and ADAMS. Instantly we can see that the ADAMS model has a slight overspeed that
appears not to be present on the Excel graph.
In general the ADAMS calculated values appear higher than those for the Excel calculations. On
consultation with peers, this seemed a common theme. Also, ADAMS suggests that the absolute Yaw
Rate will be higher with an Overdamped system than with a Critically Damped system, something that
the Excel calculations refute. In general, the Excel graphs show a greater correlation between outputs
and what we might expect from base principles.
From a physical point of view, we have seen how easily the handling characteristics of a vehicle can be
influenced by moving the CoG even a few centimetres. This analysis provides a stark insight into just
0
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6
YawRate(°/s)
Time (s)
Overdamped (ζ = 1.69)
Excel
ADAMS

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why professional racing organisations spend vast quantities of resources on optimising weight
distribution, going as far as movable ballast, when the rules permit. However, for all our theory it is
important to remain grounded:
Between our two extremes (65% weight front and 50% weight front) there is a difference of:
0.15 ∗ 906 = 136kg
Such a drastic shift in weight distribution for such a small car is remarkable and most likely involves re-
organising the engine bay, something beyond the desires and possibilities of all but the most
resourceful.

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TYRE FUNCTION
The most obvious inaccuracy with our model is that the tyres have been modelled linearly. Sadly, due to
time constraints, it was not possible to implement a full tyre model for both the ADAMS and Excel
model, but briefly we will outline the work and theory behind modelling a tyre function into ADAMS.
The function takes the form:
𝐹𝑇𝑦𝑟𝑒 = 𝑁𝐹/𝑅 ∗ 𝐷 ∗ sin(𝐵 ∗ 𝛼) − 𝐸 ∗ (𝐵 ∗ 𝛼 − atan(𝐵 ∗ 𝛼)) + 𝑆
(Pacejka, 2005)
Where 𝑁𝐹/𝑅 is the normal force acting on the front/rear, 𝛼 is the tyre slip angle and𝐵, 𝐷, 𝐸 and 𝑆 are all
scaling factors. The formula was inserted into a MATLAB script, with the values of our normal forces and
a slip angle ranging between ±12° (±
𝜋
15
rads), we manipulated the scaling factors to give us our final
formula:
𝐹𝑇𝑦𝑟𝑒 = 𝑁𝐹/𝑅 ∗ 15 ∗ sin(10 ∗ 𝛼) − 0.95 ∗ (10 ∗ 𝛼 − atan(10 ∗ 𝛼)) + 0
Which, with our initial conditions of 𝑁𝐹 = 588.9, 𝑁𝑅 = 317.1, gives us the following plot:
This is implemented in ADAMS with the following script:

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Function 2 – Tyre function for the model (LAT_F_LEAD_TYRE and LAT_F_REAR_TYRE)
Where MF/MR is the mass on the front/rear tyre respectively. Note that the functions have been
multiplied by -1 so as to work correctly in our co-ordinate system. This is calculated by measuring the
distance between the CoG and a nominal point at the rear of the body to find b:
Function 3 – Measure position of CoG relative to rear of body (POS_COG)
From this we can calculate the weight distribution ratio:
Function 4 – Calculate weight distribution (F_R_RATIO)
Finally we can find the axle loads at the front and rear:
Function 5 – Calculate front and rear normal tyre forces (MF and MR)
Thus, our model is responsive to changes in the position of the centre of gravity, though not overall
weight as to allow direct comparison of results.
The results were very interesting and the model is included (BICYCLE_TYRE). What is startling is how
sensitive the front/rear is to changes in weight distribution. With a forward CoG, the rear will easily
break away from the front if a large steering input is applied (as is to be expected from a small
hatchback at high speed). This was only an initial attempt at creating a tyre function and it is obvious
that ours is perhaps too steep and sensitive to weight transfer. This will be taken into account for any
future modelling.

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REFERENCES
Balkwill, J. (2015). Derivs I & ARB.
Milliken, W. F., & Milliken, D. L. (1995). Race Car Vehicle Dynamics.
Pacejka, H. B. (2005). Tyre and Vehicle Dynamics.

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