Tire model

Just a short post to show the tire-road contact model that I will use for the dynamical model of a bicycle that I'm developping. The model is fully "analytical", no multi-body software used. There are some secundary features lacking but the core is already done.

The GIF shows a 2kg wheel (R=0.34m) falling from 0.5m to a irregular road (maximum amplitude of irregularities=0.07m). Tire stiffness and damping are taken from previous posts of this blog. IC have been chosen randomly.

As you can see just after the impact, this road profile-wheel combination is near the limit of the single contact point condition that the majority of tire load models use. Some plots:

That's all for today. Greetings

Yaw calculator

To sum up all the posts about the topic and following advice by ST member Aralo, I have written a program that some of you will find very helpful. This program is divided in two modules, the objective of the first one (product development) is to compute the yaw-independent CdA given probability distributions or general data of bike and wind speed and relative orientation. This parameter allows you to compare different iterations of a product and make a reasonable choice based on the real world conditions that the component will face. The second module is designed to help with equipment selection for a given course and bike and wind speed. It allows to import bike position and speed data obtained from GPS tracking or any other source and compare the average power consumption needed for each setup.

The program can be downloaded from the following link. I have enclosed some data samples in the file to show the data structure that the program handles.

Yaw calculator v.1.0.

Don't forget to run MCRInstaller before running the app. Have fun and please report any bugs

Greetings

About yaw II

I wasn't going to write more posts about this subject but I've got some results to share. If you remember my previous post about yaw, I calculated the relation between the standard deviation of the Maxwellian distribution and the average of the Rayleigh distribution for wind speeds. One of the main problems of the Rayleigh distribution is that there is only one tuning parameter. That means that average value and variability can't be chosen indepently or, alternatively, that a high average wind speed means high variability.

For that reason, I have tried to modify Dan Connelly's yaw model for a more evolved wind speed model. The Weibull distribution is a two-parameter distribution that is used to model wind speed in the energy industry but its complexity means that no analytical expression can be obtained for the yaw probability model for the fixed bike speed case. If we consider that bike speed is variable, it becomes too much computing intensive and impractical.

As I wanted to take into account variable bike speeds, the Rayleigh model was the only option for wind speeds. Assuming a Weibull distribution of bike speeds, the yaw probability function for variable bike and wind speed and bike and wind direction is:

κ, shape factor of the Weibull distribution. λ, scale parameter of the Weibull distribution. va, average wind speed

The main problem of this distribution is that it isn't normalized and, as no analytical solution exists, the normalization constant can't be computed. The normalization constant could be calculated numerically but this would be too computing intensive so I have used an alternative method. If you calculate the probability for a given yaw, the real probability would be that value divided by the normalization constant. So, if you calculate the probability for different yaws and you fit that data to a null-average gaussian distribution (as Mavic's data) multiplied by that unkown constant, the standard deviation and the normalization constant can be determined.

An example. Weibull distribution for bike speeds:

Rayleigh distribution for wind speeds:

And the resulting probability function and CDF of yaw:

More soon. Thanks for reading!

Drag to time savings

In the two following sheets you can find different models to translate drag to time savings. The first model is the most complex and it leaves the door open for a pretty unintuitive result, small drag savings measured in the wind tunnel could translate to time losses in real world. I have tried to find out when this could happen but, for the moment, I haven't found any real world combination of parameters that could produce time losses.

To compare the performance of the 3 models that can be used for real world conditions (non-null wind speed), I have considered two different bike speeds for the initial run (R1). For both bike speeds, wind direction and speed have been chosen to produce a yaw equal to -10º for the initial run. Then, a CdA reduction equivalent to 50gF measured in the wind tunnel at 30mph has been applied and time savings are calculated for each model. For the model 1, I have considered Speed Concept drag-yaw linear relation (DZ on, Fastest bottle configuration) taken from Cervélo P4 in the Tunnel Slowtwitch's article. Yaw angle is considered negative when the apparent wind speed sees the driveside first. Here are the results:

As you can see, the phenomenon that causes the paradox that I have already commented is present here. Drag savings in real world are smaller than in the wind tunnel because the speed increase causes a yaw reduction and, consequently, a small drag increase. The error is maximized for yaws near stall angle, when the derivative of drag with respect to yaw is bigger.

The previous picture shows the formula that's typically used to convert from drag to time savings. As you can see, it's pretty accurate for most situations, especially when bike speed is similar to wind tunnel speed. Drag-time relation is pretty linear for all the models that I've presented so it's also accurate to predict time savings for bigger drag reductions.

That's all for today. Thanks for reading!

Drag to time savings. An introduction

As an introduction to this topic, two plots that show the relation between time savings, CdA reduction and previous CdA for a given bike speed. These two plots are obtained for the only situation where time savings are independent of wind speed and direction, the null wind speed case. The first plot shows time savings when the variations of friction and gravity power due to the speed increase are neglected. The second plot also considers these variations

As you can see, when the variations of power due to gravity and friction are neglected, time savings are overestimated. In the following posts, I will present more complex models that consider wind speed, direction and the modification of yaw angle and, consequently, CdA.

More soon. Thanks for reading!

A tool to determine airfoil shape using rim cross section

Just a small tool to see how the airfoil shape of a wheel is modified when the distance from the ground to an horizontal cut plane is changed. As the wind is constrained to flow parallel to the ground, this airfoil shape is what the wind sees depending on the distance to the ground. Bontrager commented some time ago that they tried to improve the performance of their airfoil shape for h=0 (h is the distance from the hub axle to the horizontal cut plane) because the aspect ratio is the lowest and this was the worst case scenario. I haven't got neither CFD nor wind tunnel data to back this up but, as you will see in the following graphics, chord length and the shape of the airfoil changes notably when h is modified. To illustrate how this tool works, I will use the following rim cross section:

NACA 0030 (3.333:1) symmetric airfoil with an overlaid 20mm tire. This could be assimilated to a 70-75mm rim with tall blades and curved brake walls. 27mm at widest point

The following plot shows the tire leading side for some values of h:

Wheel radius, 0.34m. Yellow, h=0.1m. Green, h=0.21m. Blue, h=0.26m. Black, h=0.32m. As you can see, when h increases, the widest point approaches to the leading edge of the airfoil. When h>wheel radius-chord length, tire leading and rim leading sides form an unique airfoil

Finally, in the following graphic, the relation between aspect ratio and h for the interval with separated tire leading and rim leading sides airfoils is shown :

It goes from 3.333:1 to 8.535:1. Average is 4.069:1

That's all for today. Greetings

A yaw-independent quantification of aerodynamic performance

The relation between yaw and aerodynamic drag makes it difficult to discern which component is really faster. Wheel manufacturers usually use sentences like "X wheel is Y seconds faster than Z wheel  over 40km" but how do they calculate that time gain if wind and bike speed is variable as well as drag as a function of yaw? The answer is using a weighting function that multiplies the drag function obtained in the wind tunnel by a factor to obtain an unique drag value.

Until now, the weighting functions that every manufacturer use haven't been known but Mavic showed his during the presentation of the Mavic CXR80. In the previous post, I worked that CDF function to obtain a normal distribution that can be used as a weighting function. Next step is to consider the drag function obtained in the wind tunnel. Typically, aerodynamic drag is considered to vary linearly between measurement points. Another aspect that has to be considered is that aparent wind speed has been variable for that weighting law but, as the fluctuations of the Reynolds numbers are small, CdA can be considered independent for a given yaw.

Let's consider a drag function with n divisions, the first abscise is called x0 and the last one, xn. The CdA for every abscise is called di, with i going from 0 to n. The weighting fuction is assumed to be a null-mean normal distribution as Mavic has showed. Now, we can define the weighed yaw-independent CdA as:

erf, error function. sigma, standard deviation of the weighting function. The leading term (before the summation) takes into account that the drag function isn't defined in the whole range of possible yaw values

Let's make an example. Consider the following drag function taken from Slowtwitch's Cervélo P4 in the Tunnel article:

As you can see, the Cervélo P4 and the Trek Speed Concept are nearly tied. Considering that wind tunnel speed was set to 30mph, approximated CdA values are:

Using Mavic's weighting function with a standard deviation of 10.3388º, the terms of the summation over each interval are the following ones:

The leading term is equal to 1.056, the closer this term is to the unit, the surer we are of the conclusion. Finally, the yaw-independent CdA values of both bikes are:

The conclusion is: "A riderless Cervélo P4 "ridden" in typical conditions will produce 100*(0.003868-0.039685)/0.03868=2.598% less drag than a Trek Speed Concept with a confidence of the 100/1.056=94.697%". As the correlation between wind tunnel data and real world data has been already proved, this method, when coupled with pedaling-rider wind tunnel data, can answer the question "Which setup is fastest?" without any doubt. 

That's all for today. Thanks for reading!