University of Notre Dame
Department of Aerospace and Mechanical Engineering
Date: February 28, 2008
To: Prof. Stephen Batill and Prof. Michael Stanisic, AME 40463 Management
From: Matt Morris, Hazardous Material Solutions, Inc.
Subject: Individual Engineering Trade Study
I. Introduction
A properly functioning drive system is an important feature of the prototype being
constructed to demonstrate the concepts behind the ‘Hound Dog’, a hazardous material search
and identify vehicle created by HMS, Inc. The drive system will be responsible for the physical
aspects of moving the vehicle and will be critical to successfully demonstrating the concepts
feasibility. A principal component of the drive system that must be analyzed to create a
successful prototype is the electric motors that will power this drive system. The purpose of this
trade study is to determine the angular velocity, torque, and power requirements for the pair of
electric DC motors that will be used to power the vehicle. These motor requirements will be
determined based on the vehicle’s velocity, acceleration, wheel size, mass, and applicable
frictional forces. The results of this study will be utilized in the process of selecting the motors
used in the drive system for the prototype.
II. Information Sources
This trade study was conducted using multiple sources of information. Sources used
include two text books, results from a trade study conducted by a member of HMS, Inc., various
internet sources, and an information packet on DC electric motors. Appendix 1 contains a list of
reference consulted through out the process of conducting this trade study.
The text books used in this trade study were Engineering Mechanics Statics and
Fundamentals of Machine Elements. These books provided insight into creating a free body
diagram for the forces acting upon a moving tire, modeling forces using torque and moment
equations, a brief overview of ball bearing interactions, and an in depth look into the interaction
of rolling friction.
The value for the coefficient of static friction used in this trade study was provided by
HMS, Inc. member Matt Prygoski. As part of his trade study, he conducted an experiment on
tire rubber to determine the coefficient of static friction. From this experiment he was able to
determine the coefficient of static friction of a tire on various surfaces. The minimum value
calculated is used in this trade study and occurred for contact between the tire and dirt.
Various internet sources were used to determine the coefficient of friction for ball bearing
interactions. Although the reliability of internet sources should be judged with caution, the
sources used were either manufactures or technical papers. The coefficients of friction for ball
bearing interactions were taken off of the websites of SKN and INA, two of the leading
manufacturers of ball bearings.
Finally, a brochure created by maxon motors was used to determine relationships
between different aspects of electric DC motors. This brochure can be found on the internet, but
was supplied for use by Gregory Brownell, the Microprocessor Technician at the University of
Notre Dame.
III. Design Variables, Design Parameters, and State Variables.
This trade study can be broken down into Design Variables (DV), Design Parameters
(DP), and State Variables (SV). The Design Variables help to define a specific instantiation of a
design, while the Design Parameter help to specify a feature of the design or the environment in
which the design functions. The State Variables are a result of analysis of the Design Variables
and Design Parameters and are used to describe the performance of a specific design. The DV,
DP, and SV that are part of this study can be found in Table 1.
Design Variables
Design Parameters
State Variables
Linear Velocity (m/s)
Radius of Tire (in.)
Angular Velocity (rpm)
Linear Acceleration (m/s2)
Radius of Motor Shaft (in.)
Torque (N-m)
Vehicle Mass (kg)
Coefficient Static Friction
Power (W)
Coefficient of Friction in Bearings
Table 1: Variables Used in Trade Study
The DV for this trade study were chosen because they are performance requirements
that can be set for the prototype. The DP for this trade study can be broken down into two
groups. The first category consists of DP that are a results of design decisions and includes the
radius of the tire, the radius of the motor shaft, and the coefficient of friction in the ball bearings.
The second category consists of DP that are a result of the environment and includes the
coefficient of static friction. The SV for this trade study were chosen because their values are
directly influenced by manipulation of the DV and can be useful in choosing the motors to power
the prototype.
IV. Constraints
The constraints on the DV are a consequence of keeping the prototype as simple as
possible. The constraints on each of the three DV can be viewed below in Table 3. The
minimum constraint on the linear velocity is due to the fact that the vehicle must be able to carry
out a search for hazardous material in a swift and reasonable time. The maximum constraint on
the velocity is in place so that the vehicle travels at speeds that do not exceed the rate of
detection. The reasoning behind the minimum constraint on the linear acceleration is the same as
the reasoning behind the minimum constraint on velocity. The maximum constraint on
acceleration is a consequence of the desire to avoid tire slip during acceleration. As long as the
acceleration stays below the maximum value in Table 2, it can be assumed that the tire will not
slip. Finally, the minimum constraint on vehicular mass is a rough estimate of the smallest
vehicle size, while the maximum constraint is a rough estimate of the largest size the vehicle can
have before it becomes too heavy to be moved by one person.
Design Variables
Minimum Constraint
Maximum Constraint
Linear Velocity (m/s)
2 m/s
5 m/s
Linear Acceleration (m/s2)
1.5 m/s2
3 m/s2
Vehicle Mass (kg)
5 kg
25 kg
Table 2: Constraints on Design Variables
V. Engineering Analysis
To begin this trade study, research was conducted on how electric DC motors work. The
source for this information was the maxon motors brochure Key Information on maxon DC
Motor and maxon EC Motor. DC electric motors convert electric power in the form of current
and voltage into mechanical power in the forms of speed and torque. The conversion from
electrical power to mechanical power is not 100% efficient and power loses occur due to
resistance and frictional with in the motor. In every DC motor, increasing the voltage increases
the power output. The speed and torque that are output as mechanical power from the motor are
inversely proportional to each other at a constant voltage. From this information on DC electric
motors, it can be concluded that the key aspect of the motor which should be focused on in this
study is the mechanical power produced by the motor.
The required mechanical power is determined from the forces acting on a tire that is
powered by the motor. A free body diagram of the forces acting on a wheel moving from right
to left can be seen in Figure 1.
Figure 1: Free Body Diagram of Forces Acting on a Tire
This free body diagram contains assumptions made to simplify the analysis of the forces acting
on the wheel. The primary assumption is that the mass of the vehicle is equally distributed
between the two real wheels, which means that each wheel is loaded with a force of mg/2. In
reality, this is not the case. However, this assumption allows for a simple calculation of load
distribution. In addition, this conclusion leads to the assumption that negligible forces are
exerted by the casters at the front of the vehicle because they carry no load. A secondary
assumption made in this free body diagram is that the drag force exerted on the vehicle is
negligible. The drag force, Fd, on a vehicle is calculated using the equation:
 V2
Fd  Cd A a .
(1)
2
In this equation Cd is the drag coefficient (which is between 0.35 and 0.5), A is the crosssectional area at the front of the vehicle, ρa is the density of air, and V is the vehicle’s velocity.
This assumption can be made because the vehicle’s maximum velocity as defined in this trade
study is not high enough to really cause significant drag forces.
From the free body diagram seen in Figure 1, an equation can be derived that indicates
the torque needed to move the vehicle. The equation is derived under the assumption of a no-slip
condition between the tire and the surface over which the wheels are traveling, the contact
surface. Assuming no-slip in the tire indicates that the tire is stationary at point A, where the tire
meets the contact surface in the free body diagram. The critical aspect of the no-slip condition is
that at an instant in time, there is no angular acceleration around the center of rotation, which
means that the sum of the moments about point C in the free body diagram must be equal to zero.
This results in the following equation:
(2)
 M C  0  T  FS r  TB ,
where T is the torque produced by the motor, FS is the force of static friction, and TB is the torque
that results from friction in the bearings to which the shaft is attached. Solving this equation for
T results in an equation that gives the magnitude of torque needed to power the vehicle:
(3)
T  FS r  TB
To be useful in analysis, Equation 3 must be in terms of the DV and DP. Summation of forces in
the direction of motion will allow the determination of the force of static friction in terms of a
DV and a DP. The resulting equation is
ma
F 
 FS ,
(4)
2
Another benefit of this equation is that it can help determine the maximum acceleration under
which no-slip conditions still apply. If the vehicle acceleration exceeds the maximum static
frictional force, FS= μSN where N=mg , then the wheel slips. Therefore the max acceleration can
be determined from the maximum force of static friction. The resulting equation is
a max   S g .
(5)
Assuming no slip also results in the conclusion that there is negligible rolling friction acting on
the tire, which means that rolling friction does not need to be calculated when determining the
torque. The resultant bearing torque, TB, in terms of the DV and DP can be reduced down to
mg
TB  2  B
rB ,
(6)
2
where μB is the coefficient of friction for a ball bearing assembly and rB is the radius of the motor
shaft on which the torque acts. The coefficient of 2 is in the equation because the design calls for
the motor shaft to rest on two ball bearing assemblies. Combining Equations 3, 4, and 6
produces:
ma
mg
T
r  2 B
rB .
(7)
2
2
This equation solves for the required motor torque, a SV, in terms of the DV of mass and
acceleration, and the DP of the tire radius, motor shaft radius, and coefficient of friction for the
ball bearings.
Finally, the required mechanical power can be calculated using the torque and angular
velocity. The angular velocity is calculated using the DV of velocity and the DP of the tire
radius. The angular velocity, ω, that results from a velocity of V and a radius of r is
V

.
(8)
2r
From this value of angular velocity, the mechanical power for the motor can be calculated using
the equation
P  T ,
(9)
where power is in units of N-m/s which is commonly referred to as Watts.
V. Results
Analysis of the motor requirements was conducted using a MATLAB program that
incorporated Equations 5-9 to model the behavior of the SV based of the DV and DP. A copy of
the code is found in Appendix 2. The DV used in this study along with the constraints on these
variables can be found in Table 2. The values for the DP used can be found in Table 3 below.
Design Parameter
Value
Radius of Tire (in.)
Radius of Motor Shaft (in.)
Coefficient Static Friction
Coefficient of Friction in Bearings
4 in.
½ in.
0.7
0.01
Table 3: Design Parameter Values Used in the MATLAB Analysis
The value for the radius of the tire is based on the 8 in. diameter tires that will be used on the
prototype. The radius of the motor shaft is an estimate based on the size of the tire hub. The
coefficient of static friction is the minimum value for μS determined by an experiment conducted
by Matt Prygoski for his Trade Study. The coefficient of friction for ball bearings is μB=0.01, a
maximum value which was determined from information supplied by bearing manufacturers
SKF and INA. Results of the analysis can be seen below in Figures 2 through 4.
Figure 1: Plot of Power vs. Velocity for Various Accelerations at a Constant Mass m=15 kg
Figure 2: Plot of Torque vs. Mass for Various Accelerations at a Constant Velocity V=2.5 m/s
Figure 3: Plot of Power vs. Mass for Various Velocities at a Constant Acceleration a=2.0m/s2
Figure 1 depicts the variation in required mechanical power for different velocities at
different accelerations at a constant mass of 15 kg. From this figure, it can be concluded that
mechanical power is linearly influenced by velocity, with power increasing with an increase in
velocity. It can also be concluded that the magnitude of this increase is influenced by
acceleration since the magnitude increases with increasing acceleration.
Figure 2 depicts the variation in required torque for changes in vehicle mass at different
accelerations and a constant velocity of 2.5 m/s. From this figure, it can be concluded that torque
requirements increase for increasing vehicle mass and that the increase in torque requirements
occurs more rapidly for higher accelerations.
Figure 3 illustrates the variation in required mechanical power that occurs for different
masses at various velocities and a constant acceleration of 2.0 m/s2. This illustration leads to the
conclusion that, like torque, power requirements increase with mass. It can also be concluded
that the magnitude of this increase grows with increasing velocity.
VI. Conclusions
A few conclusions can be drawn from the three figures that were the results of analysis.
It is clear that the mass and acceleration greatly influence the required torque, while the velocity
influences the required angular velocity. It is also clear that the required mechanical power is
influenced by all three of the design variables. The design parameters that have the greatest
influence on motor requirements are the tire radius and the coefficient of static friction. When
considering the conclusions drawn from the results of this trade study, it is important to
remember that these results are hypothetical and based off of numerous assumptions and
conclusions. Therefore the results and the conclusions drawn from them might not be 100%
accurate. However, they can be very useful when utilized as guidelines for choosing the motors
to power the prototype.