ME 6105
Homework #4: Uncertainty Analysis
Mountain Bike Suspension Optimization
Neerav Patel
David Bourbon
11/11/2010
Task 2: Identify and Model the Uncertainty in your Design Analysis
Model
For mountain bike suspension design analysis model, the most important
variables were determined to be the front suspension spring and damping constants, the
rear suspension spring and damping constants, the rider’s mass on the handlebars, the
rider’s mass on the seat, the rider’s mass on the pedals, the front wheel dropping height,
the rear wheel dropping height, the initial bike velocity, and the spring and damping
constants for the tire stiffness measure. The experiment was set up to drop the bike from
a range of heights. The front and rear wheels were initialized at different heights
throughout the experiments with different initial bike velocities as well. The rider masses
and suspension characteristics were varied as well. There was some uncertainty on how
much of an effect the tire stiffness of the bike would have on the vertical motion of the
rider and the forward motion of the bike. Therefore, the spring and damping constants
that correspond to the tire stiffness were included in the experiments. Table 1 shows the
Central Composite experiment uncertain variables and the initial triangular distributions.
The experiment outputs observed were bike travel distance and velocity in the x direction
(forward motion of the bike) as well as the handlebar and seat travel heights. The bike
travel distance and velocity in the x direction correspond to the forward motion which is
one of the primary experimental outputs. The handlebar and seat travel heights represent
the rider comfort which is the other primary experimental output. The handlebars and seat
are main contact points with the rider so the motion of these points essentially represents
the motion of the rider. Using the variables and bounds shown in Table 1, a sensitivity
analysis was performed to observe the effects of the uncertain variables on the desired
outputs.
Table 1: Uncertain Variables and Initial Triangular Distributions
Lower
Upper
Expected
Variable
Units
Bound
Bound
Value
Front Spring Constant
N/m
1000
10000
5500
Front Damping Constant
Ns/m
100
1000
550
Rear Spring Constant
N/m
1000
10000
5500
Rear Damping Constant
Ns/m
100
1000
550
Front Wheel Drop Height
m
0
2
1
Rear Wheel Drop Height
m
0
2
1
Rider Mass on Handlebars
kg
10
20
15
Rider Mass on Seat
kg
5
10
7.5
Rider Mass on Pedals
kg
40
70
55
Tire Spring Constant
N/m
1000
10000
5500
Tire Damping Constant
Ns/m
100
1000
550
Initial Wheel Velocity
rad/s
0
60
30
The forward motion of the bike was observed in the experiment from the overall
distance of travel in the x direction and the bike velocity. The velocity was only
initialized and no additional velocity inputs were used during the experiment. This is
representative of “stopping pedaling” on a trail when the bike is about to reach an
obstacle. Figure 1 shows the main effects of the variables on the travel distance while
Figure 2 shows the main effects of the variables on the bike velocity in the x direction.
For the bike travel distance, initial angular velocity of the bike wheels had the largest
effect on the distance. This is makes intuitive sense because it is the main factor in
determining the rate of change of the distance. The three mass contact points had an
effect on the distance. The mass on the seat, or directly above the swingarm pivot, was
the smallest of the masses due to the rider often standing up during mountain bike riding.
Thus, it makes sense that this effect is small. The majority of the rider mass is on the
pedals and handlebars which manifested in the Central Composite experiment. The
results are somewhat questionable with regards to the suspension of the bike. The front
wheel and suspension spring constants showed that they influenced the distance but the
rear wheel and suspension constants, front wheel and suspension spring constants, and
the drop heights showed no effects. This is most likely a result of how the model was
built. The frictional effect of the ground on the tires is essentially the main obstacle for
forward motion. Thus, increases in masses influence the results. However, the spring and
damping characteristics do not make a large difference. The main effects of the variables
on the bike velocity (Figure 2) show the same trends as the effects on the travel distance.
However, the masses on the handlebars and the pedals had a higher impact but still not as
high as the initial angular wheel velocity. In order to examine the forces exerted on the
rider, the force on the handlebars was added to the experiment. This is not only
representative of rider comfort, but in reality, would have an effect on the bike speed due
to the riders pedaling and braking response.
InitAngVel
54%
massPedals
17%
massFrontForks
14%
FrontWheelConst
8%
FrontSpringConst
5%
massRearSw ingarm
1%
RearSpringConst
0%
RearDampConst
0%
FrontDampConst
0%
FrontWheelDamp
0%
DropHeightFront
0%
DropHeightRear
0%
0
5
10
15
20
25
30
35
40
45
50
55
60
65
Figure 1: Bike Travel Distance
70
75
80
85
90
95 100
InitAngVel
37%
massPedals
23%
massFrontForks
19%
FrontWheelConst
10%
FrontSpringConst
7%
massRearSw ingarm
3%
RearSpringConst
1%
RearDampConst
0%
FrontDampConst
0%
FrontWheelDamp
0%
DropHeightFront
0%
DropHeightRear
0%
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95 100
Figure 2: Bike Travel Velocity
As previously discussed, the force on the handlebars was added to the Central
Composite experiment in order to examine the effects of the variables on the force
exerted on the rider. The results were interesting in that all of the variables except for the
drop heights show some effect on the force on the handlebars. It makes sense that all of
the suspension and tire stiffness characteristics would influence the impact force.
However, the drop height should have an effect on the force. Higher drop heights should
result in higher impact forces.
FrontWheelConst
30%
massPedals
18%
FrontDampConst
11%
RearSpringConst
10%
FrontSpringConst
8%
massRearSw ingarm
8%
InitAngVel
6%
RearDampConst
4%
massFrontForks
3%
FrontWheelDamp
2%
DropHeightFront
0%
DropHeightRear
0%
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95 100
Figure 3: Vertical Force on Handlebars
In addition to the vertical force on the handlebars, the magnitude of vertical
motion on the two main rider-bike contact points can be used to examine the rider
comfort level. The motion of the handlebars and the pedals (Figures 4 and 5,
respectively) showed nearly the exact same results from the Central Composite
experiment. The results and interpretations are similar to the results for the x direction
motion of the bike. However, the initial angular velocity showed no effect on the vertical
motion. It appears that the most uncertainty results from the initial wheel angular
velocity, the mass on the pedals, the mass on the handlebars, and the front wheel and
suspension spring constants.
massFrontForks
34%
FrontSpringConst
22%
FrontWheelConst
22%
massPedals
18%
massRearSw ingarm
3%
RearSpringConst
1%
RearDampConst
0%
InitAngVel
0%
FrontWheelDamp
0%
FrontDampConst
0%
DropHeightFront
0%
DropHeightRear
0%
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95 100
70
75
80
85
90
95 100
Figure 4: Handlebar Travel Height
massFrontForks
33%
FrontSpringConst
23%
FrontWheelConst
23%
massPedals
17%
massRearSw ingarm
3%
RearSpringConst
1%
RearDampConst
0%
InitAngVel
0%
FrontWheelDamp
0%
FrontDampConst
0%
DropHeightFront
0%
DropHeightRear
0%
0
5
10
15
20
25
30
35
40
45
50
55
60
Figure 5: Seat Travel Height
65
Task 3: Elicit a Detailed CDF for the Most Significant Uncertain
Variables
As discussed in the previous section, the most uncertain variables from the
Central Composite experiment were the initial wheel angular velocity, the rider mass on
the pedals, the rider mass on the handlebars, the front tire spring constant (stiffness), and
the front suspension spring constant.
The CDF for the initial wheel angular velocity is shown in Figure 6 below. The
beginning of the elicitation process involved asking what bike velocities are commonly
achieved by riders. Most bikes are ridden under 8.94 m/s (about 20 mph). Thus, we can
say that there is a probability of 0.9 that the rider will be under this speed which translates
to about 27 rad/s with 0.66 m diameter wheels. It is assumed that all bikes are ridden at
below 19.8 m/s and above 1.65 m/s which correspond to 60 rad/s and 5 rad/s wheel
angular velocities, respectively. When asking where the halfway break point was in the
range of angular velocities, it was determined that a speed of 4.62 m/s corresponding to
about 14 rad/s was reasonable. Therefore, about half of the riders ride below 4.62 m/s
while half ride above 4.62 m/s. The resulting CDF is reasonable using the elicitation
process detailed. It is steep early in the distribution and levels off. This corresponds to the
majority of riders riding at the lower end of the speeds in the range.
Figure 6: CDF for the Initial Angular Velocity
The mass of the rider was split between the pedals, the handlebars, and the seat.
The majority of the mass was on the pedals and the handlebars. Figure 7 shows the CDF
for the mass of the rider on the pedals. First, the distribution of the rider’s weight was
elicited. It was determined that about 70% of the weight should be on the pedals while
20% is on the handlebars and 10% is on the seat. This was determined by examining
normal mountain bike trail riding conditions. Most often, the rider will be upright on the
bicycle. With 70% of a normal adult’s weight on the pedals, upper and lower bounds
were elicited by asking what the most and least a normal adult would weigh. Then, the
upper and lower bounds were taken as being 70% of those values. This resulted in 40 kg
and 70 kg as the bounding range. The average value of 55 kg was used as being a
reasonable halfway point so it was given a probability of 0.5. It is less likely that the rider
would be near the bounds so probabilities of 0.1 for a 45 kg mass and 0.9 for a 65 kg
mass were added. The same elicitation process was used to determine the CDF for the
mass of the rider on the handlebars. This CDF can be seen in Figure 8. Both CDFs make
intuitive sense. They result in distributions for their probability density functions which
are somewhat similar to the normal distribution.
Figure 7: CDF for the Rider Mass on the Pedals
Figure 8: CDF for the Rider Mass on the Handlebars
The spring constant from the front forks directly impacts the rider. This force goes
directly to the rider in such a way that if there is too little or too much of a spring effect, it
will cause the rider to let go of the handlebars and potentially crash. Using Hooke’s Law
(F=-kx), a mass on the handlebars of 100kg, and a travel distance of 0.1m [Ref. 1], the
spring constant ends up to be around 10,000 N/m. With a weight of 10kg, this constant
reduces to around 1,000N/m. Typically, when traveling on a bike, a person would exert
around 20kg when traveling on an incline. This causes the CDF to increase rapidly after
reaching a spring constant of 2,000 N/m.
Figure 9: CDF for the Spring Constant affecting the Front Forks
Typical values for spring constants on a front tire are around 6000N/m. The value
was found from the equation k=mg/r where k is the spring constant, m is the mass in
kilograms, g is the acceleration due to gravity, and r is the radius of the bike. This is an
average value so anything above it would be more than what the rider would need. In
colder climates, less damping force is needed to obtain better grip but considering most
of the time the trails are ridden during temps of 50 to 100 degrees, this underinflation is
not used often except in expert or professional courses. This is why the CDF decreases
quickly.
Figure 10: CDF for Spring Constant affecting Front Wheel
Task 4: Determine the Distribution of the Output of your Model
To determine the distribution of the model, the CDFs were used in conjunction
with mountain bike Modelica model and the Monte Carlo plug-in. Figure 11 shows the
GUI for the Monte Carlo plug-in and Figure 12 shows the layout of the ModelCenter file.
The values used for the triangular distribution were found from Table 1. A uniform
distribution ranging from 0 to 1 was used for the variables in which detailed CDFs were
elicited. Then, the Excel CDFs were used over this range.
Figure 11: Probabilistic Analysis GUI
Figure 12: Model Center Model
Using the model from Figure 12, a Monte-Carlo analysis was used to create
histograms for each of the same outputs as in Task 2. Figure 13 shows the histogram for
the force on the handlebars. The histogram shows that the results are skewed to the right
and there is a small standard deviation. However, the numerical values seem to be nearly
the same between the expected value and median. Figure 14 shows the histogram for the
seat travel height during the drop test. The shape is similar to that of a normal
distribution. However, there is an additional peak to the left of the expected value. Figure
15 shows the histogram for velocity in the x direction of the Dymola simulation. The
velocity analysis results are also skewed slightly to the right, but the distribution is much
smoother than that of the force on the handlebars. The expected value is larger than the
median value. Figure 16 shows the histogram for the handlebar travel height during the
drop test. This histogram is skewed slightly to the left meaning the expected value is less
than the median. Figure 17 shows the histogram for the total distance traveled by the bike
in the x direction. The histogram for the total travel distance is skewed to the right as
well. Figure 13 for the force on the handlebars should be more representative of a normal
distribution. The extreme right skew could be a result of the lower and upper bound
selections for the triangular distributions. Again, the numerical values hardly change over
the distribution so the results are basically concentrated on one location. Figures 15 and
17 are nearly identical which make sense because they are affected by the variables in
nearly the same way as shown in Task 2. These two forward motion histograms are
skewed to the right. This could be due to the experimental setup. The bike drops and is
allowed to bounce. Further into the simulation, the bike more frequently contacts the
ground and creates forward motion. Figures 14 and 16 for the seat and handlebar
bouncing height, respectively, are nearly identical as well. This makes sense because they
are outputs of nearly the same variables. These values should be slightly skewed to the
left because the bike is initialized at a height and then dropped such that gravity
eventually stabilizes it into contact with the ground. Higher heights are achieved early in
the simulation.
Figure 13: Histogram for Force on Handlebars
Figure 14: Histogram for Seat Bounce Height
Figure 15: Histogram for Velocity in X Direction
Figure 16: Histogram for Handlebar Bounce Height
Figure 17: Histogram for Distance Traveled
Task 5: Lesson Learned
Neerav Patel
I feel that I learned about the uncertainty of all modeling and simulations with this
assignment. Particularly, I learned a lot through eliciting CDFs for the uncertain
variables. The elicitation process was interesting and more subjective than I had
originally thought. Also, the results of Task 2 were not like I had expected. I thought the
results would be much “cleaner” than they actually were. Instead, the effects of the
variables were much more evenly split and, as a result, more difficult to decipher. This
may have been a result of the Dymola model used and I wish there had been more time to
develop and debug the Dymola model. I also learned the importance of selecting ranges
and step sizes for varying variables. This has a large effect on computational time and
accuracy of the results. I think, at times, we became too conservative in the initial
selection of lower and upper bounds for the experiment variables. The experiment times
in ModelCenter were also surprising. We conducted the experiments overnight to use our
time efficiently. I learned the importance of planning for such computationally expensive
tasks.
David Bourbon
This part of the project included some difficulties which were not seen and in
hindsight I am glad we started early. The simulations for task 2 took a significant amount
of time. This portion took close to 7 to 8 hours to complete. Luckily I have a computer
with a great deal of processing power (quad core with 4 gigs of ram) which probably cut
down the time it took to run this as opposed to a laptop. From this I learned why
companies spend so much money on the hardware that their employees use. It would be
crazy to think that researchers would have to wait an entire work day to finish simulating
a problem before it can be analyzed. Luckily once this part was done, the rest of the
assignment did not take nearly as long. Another thing learned was also how careful we
have to be when using two different computing programs. This required double and triple
checks to make sure that what was being inputted would not break and cause a lot of time
to be wasted.
References
1. “Top Fuel 9.9 SSL” Trek Bikes.
http://www.trekbikes.com/us/en/bikes/mountain_full_suspension/top_fuel/topfuel
99ssl/