Some help calculating yaw-weighted CdA values

Hello everybody,

Short post but many of you will find it quite useful. As the design of bicycle components that are better tailored for the real-world conditions that the riders face has become more common, the concept of yaw-weighted CdA has gained popularity. For those that don't know about the term, yaw-weighted CdA can be understood as a mean CdA value representative of the conditions you ride in (under the assumption that Reynolds number effects are negligible) that is computed by weighting the WT/CFD yaw sweeps by the real-world yaw angle probability distribution.

Probably one of companies that has done more for the popular understanding of this metric is FLO Cycling. They wrote a first post on the topic when they launched their first generation of wheels (they employed a somewhat strange weighting function that was way off if we consider more recent real-world yaw angle probability distribution measurements) and a more mature second one where they explained a bit more in detail their methodology and employed the real-world yaw measurements that they used to drive the development of their second generation of wheels.

The method that they explain in that second post is a bit cumbersome if you want to do it yourself so I have decided to share with you a simple spreadsheet that I have been using for quite a long time. Hopefully this will simplify the task of computing yaw-weighted CdA values and make this metric even more widely used.

The spreadsheet gives the weights to be multiplied to the CdA values at the different yaw angles in order to compute the yaw-weighted CdA values. The method is based on a post that I wrote nearly four years ago. I have assumed that the component is symmetrical (if it is not you just have to use half the weight for each of the CdA values at + yaw angle and - yaw angle) and that the yaw-angle probability distribution is a null-mean Gaussian (as common sense indicates and many have found out experimentally). The procedure involves weighting the CFD/WT CdA data (assumed to be a piecewise linear fuction) by the Gaussian distribution. I have done this for a variety of values of the standard deviation/average absolute value of yaw (abs(yaw)). From this, you can modify the average abs(yaw) values, add/remove/change the yaw angles where you have data, etc. The only thing worth noting is that, as the CdA values are not measured in the full [-180º,180º] yaw range, I have assigned the remaining weight to the last considered yaw angle (20º in the original version) so you will need to move the formula in that column if the number of considered yaw angles is different.   

If you don't know what average abs(yaw) value to use, I have analysed previously published real-world measured data and these are some typical values:

FLO: 5.67º
SwissSide: 3.39º
Mavic: 8.25º

Using this and the ROT 0.005 m^2 CdA reduction=0.5 s/km time savings you can better estimate the time savings that you will observe in real-world conditions.

Playing with the limits of UCI rules

Hello everybody,

Long time without posting (18 months), I lost a bit of motivation but happy to see that people still come here quite regularly looking for updates. Many things have passed since then: finished my undergraduate studies in France and Spain, went to Brazil for my final year internship and currently pursuing a MSc in CFD at Cranfield University, UK.

This post is related to the UCI rules on equipment and how subtle they can be. What happened a few years ago in F1 with front wings that employed aeroelastic phenomena in order to improve performance is a good example of what happens when rulebooks are too detailed and does not give room for too much improvement using "traditional" methods. Engineers always find ways to improve performance and the governing body introduce more complex rules (deflection tests for front wings) that benefit nobody. Hopefully the same will not happen for bikes.

With this I don't want to be critical with the current version of the UCI rules in any way. I think that it is good to have a document that defines the rules of the game. Just noting how open to subjectivity are some of these rules and the problems that may arise if more manufacturers try to play with the limits that they set. Latest version of UCI's clarification to the rules can be found here.

The latest road "super" bikes, the Specialized Venge VIAS and the Trek Madone 2016, are a good example of pushing the boundaries of what the UCI rules allow.

First example, the front brake of the Venge VIAS. First, let's see what the UCI rules say about brake-fork integration.

Now take a look to the front end of the Venge VIAS and determine the width of the bounding box that contains the fork and brake.

Ok, it is clear that the fork and brake does not fit within a 8cm box. From the UCI rules, a brake is considered to be standard if "their shape and system of attachment allow them to be used on all types of frames and forks." That is clearly not the case for that front brake. By elimination the brake should be cosidered "integrated", shouldn't it? And this means that the brake "...whether a cover is fitted or not,... must in all cases be contained within the corresponding 8cm box". Let's read the interpretation of an "integrated" brake and imagine how it could be legal: "...which are designed for a specific model of frame/fork and which can only be used with this frame/fork due to their shape or attachment system". So if you build two minimally different models of fork and frame that use the same integrated brake, you are free to go because your integrated brake does not fit within their definition of an "integrated" brake and the dimension limits do not apply. This has not happened yet but it may have been taken into account when considering if the bike is legal.

So, if this brake is legal by not being compatible with UCI's definition of "integrated", why the typical brakes integrated in the fork (e.g. BMC TMR01) cannot use the same argument and go beyond the 8cm box? The reason is that in the UCI rules there is a specific paragraph limiting the dimensions of brakes with covers: "The combination of the frame tube (or fork tube) + brake + cover must respect the 1:3 rule, as well as the minimum and maximum dimension rules and must be contained completely within the corresponding 8 cm box.".

So, Specialized basically found a loophole in the rules by playing with the definition of an "integrated" brake and not using a cover. Hats off.

Second example, the seat tube of the Madone 2016. First, let's read the UCI rules about fairings:

"Any device, added or blended into the structure, that is destined to decrease, or which has the effect of decreasing, resistance to air penetration or artificially to accelerate propulsion, such as a protective screen, fuselage form fairing or the like, shall be prohibited."

Now, let's take a look to the internal structure of the frame.

From Madone's white paper.

Inner seat tube. From this video. Note the stepped joints (patent).

Outer seat tube.

The whole system.

The seat tube structure is basically the same as their comfort-oriented Domane except that the Madone has an aerodynamic outer tube from the BB to the seat stays-TT junction. Now remember the UCI rules about fairings. Is the outer tube really necessary or is it just a fairing for the inner one? You can argue that the inner one is not enough to give enough torsional stiffness at the BB. But, is it really the case if you are already producing the Domane and it is stiff enough? I suppose that Trek has given convincing answers to these questions when passing the approval procedure.

Some interesting questions arise: will the UCI need to use the famous motor-detecting scanners to study the internal structure of the frames when conducting the approval procedure to know if a tube is structural or just a fairing? will the approval procedure include deflection tests in the future (like for the F1 wings) in order to check this?

A very interesting topic. I would love to hear your opinions.

Eduardo Bueno

A deeper look at bicycle stability and maneuverability

Now that I've got some free time, I think that I should explain better the "A minimal bike geometry" post. In that post I said that dynamical behaviour of a bike can be defined using just 4 parameters ("modified" trail, chainstay length, wheelbase and BB offset). One might argue that such a definition of bike geometry is arbitrary but it steams from a deep analysis of steering and bike balance mechanics.

I haven't seen yet a complete enough analysis/explanation of the relation between bike geometry parameters, balance mechanics and rider input so this will be my aim in the following lines. As a first step, let's consider the typical 4 body (two wheels, frame and fork) singletrack vehicle model. Assuming RWS, this model has 3 DOF tipically defined as the rotation of the rear wheel, the roll angle of the frame and the steering angle.

At this moment, it's interesting to present the rider as both a controller (due to his capability to control the steering motion and displace his weight laterally to control the rolling motion) and a disturber of the balance (due to the pedalling loads). The controlling actions of the rider induce loads and moments in the wheels that tend to stabilize the whole system that has certain inertial characteristics.

Once this is clear, let's pass to analyze the two basic states of motion of the bike: rectilinear motion while pedaling and turning without pedaling. Motion while pedaling standing is more obvious but it's also present when pedaling seated.

Pedaling motion

1- Starting with the crankset horizontal and the left leg doing the downstroke movement, the pedaling loads generate a moment that makes the bike roll to the left. To compensate this, the rider applies higher vertical forces to the left side of the handlebar and steers to the right. Due to this steering moment and as the system tends to decrease its potential energy, a clockwise pitch motion that is only possible if the bike rolls to the right starts. As a result, the COG of the system displaces to the right and the gravity induces a moment contrary to the one generated by pedaling loads.

Loads:
- Roll motion. Encouraging desired motion: moment due to gravity and handlebar loads. Against desired motion: moment due to pedaling loads, gyroscopic moments, overturning moment in both wheels and centrifugal forces due to the spiral motion of the COG
- Steering motion. Encouraging desired motion: steering moment. Against desired motion. steering moment due to the product of lateral forces in the tire and trail (variable).

2- As the left leg approaches its lowest position, the roll angle reaches its highest value (to the right) and the steering angle is null. The pedaling loads become smaller and the rider starts to steer the bike to the left to reduce the rolling moment generated by gravity, causing a counterclockwise pitch motion that increases the height of the COG of the system. In this moment, steering and roll angles have different sign, something that reduces the leverage of the rolling moment due to gravity so the roll angle starts to decrease helped by handlebar loads. When the roll angle is 0º, the steering angle is maximum and the cycle starts again.

Loads:
- Roll motion. Encouraging desired motion: moment due to pedaling loads, overturning moment in both wheels and moment due to handlebar loads. Against desired motion: moment due to gravity, gyroscopic moments and centrifugal forces.
- Steering motion. Encouraging desired motion: steering moment. Against desired motion: steering moment due to the product of lateral forces in the tire and trail (variable).

The followings plot will help you to understand how it works. In the first one, cadence is constant and equal to 90rpm and roll and steering angles are assumed to be sinusoidal. As you can see, roll and steering angle are π/2 out of phase.

This one shows the relation between pitch and roll and steering angles for a typical bike. As you can see, roll and steering angle naturally tend to have the same value to minimize potential energy.

Turning without pedaling

In the case of turning without pedaling, the rider starts by countersteering to induce gyroscopic, steering (due to the lateral forces in the contact patch) and overturning moments that tries to turn the bike in the desired direction. When the rider has defined his desired trajectory and calculated the path radius needed, he modifies steering and roll angles in order to 1) ensure that the path of the wheels is the desired 2) turn in steady state conditions (moments due to gravity equal to moments due to centrifugal forces). Once the bike approaches the exit of the curve, the rider countersteers again inducing a counterclockwise pitching motion that increase the height of the COG. Thanks to the same mechanism previously explained in the second part of the motion while pedaling, the roll angle tends to zero and the bike return to the vertical position

Once we have analyzed the way the bike is controlled, we can define the parameters that define the manouverability and stability of a bike. Leaving the influence of the components aside, these are the following ones:

1) The vertical position of the COG of the system. This parameter modifies the rolling moment generated by the gravity for a given roll angle. It's just a function of the BB offset for a given rider.

2) The weight distribution between both wheels. Steering moment is proportional to the product between front wheel loads and "modified" trail. Weight distribution is a function of chainstay length and wheelbase for a given rider.

3) Head tube angle (HTA). The axis of precession of the wheel as a gyroscope is the axis of the steering column so HTA has an effect on gyroscopic moments.
4) How camber of the front wheel changes when roll and steering angles are modified. Wheel loads, specially overturning moment, depend on camber.

5) The relation between path radius and roll and steering angles. In short, how much you need to steer and make the bike roll for a given radius of a curve.
6) The relation between "modified" trail and roll and steering angles. This the leverage of steering moments due to loads in the wheel-ground contact.

7) The relation between frame pitch and roll and steering angles. The most obvious effect of this can be seen during the turn before a final sprint. Riders usually lower their upper body to minimize the rolling moment due to centrifugal loads. If a bike pitches clockwise more than other for a given path radius (or, alternatively, for given steering and roll angles), COG height decreases and the rider can make the turn faster. I think that this is well related to what is tipically called "a bike that just wants to go" or "aggresivity in the corners".

From 1-2 we can say that BB offset, chainstay length and wheelbase are needed to define the behaviour of the bike.

About 3), it has been said many times that gyroscopic moments are very important for the stability of a bike but it has been shown both analitically and experimentally (with a counter-rotating wheel) that they have much lower influence than gravity-induced ones. We can then neglect the influence of head tube angle for now.

Regarding 4), camber generates an overturning moment that affects the rolling motion of the bike. Camber of the front wheel for a given combination of steering and roll angles is affected by head tube angle but is it significative compared to the rolling moment due to gravity? To check this, I have plotted the ratio between rolling moments due to gravity and total (both wheels) overturning moment assuming a small steering angle. As you can see, the contribution of overturning moments to total rolling moments is negligeable so the role of the head tube angle can be neglected for the moment.

800N rider+bike. 25c tire. Considering lateral force significatively smaller than vertical force.

For 5-7, I could have developed expressions for these relations using a 4-body model but I have consulted the very interesting Motorcycle Dynamics where they are already derived. Apart from the parameters that I have already defined as necessary (BB offset, chainstay length and wheelbase), there are only two that have an influence on these relations: "modified" trail and head tube angle (HTA). To check how these dimensions affects parameters 5-7, I will define a typical bike. The geometric parameters of the typical bike are the following ones:

Regarding 5), path radius is mainly a function of steering angle and it's independent of "modified" trail. The very strong dependence on steering angle makes that important changes in the HTA doesn't affect significatively the way the bike handles because the rider can compensate those modifications by changes of the steering angle of the order of thousandths of degree. As you can see in the following plots, path radius changes due HTA are very small compared to those caused by steering motion so we can say that path radius is nearly independent of HTA.

About 6), "modified" trail is the most important parameter that affects steering motion. Although trail is defined when both roll and steering angles are null, it isn't constant and changes due to those two rider inputs. To check the influence of HTA and "modified" trail at standstill in the evolution of this parameter, I have done a small sensitivity study by changing baseline values of each parameter by 6%. As you can see in the following plots, "modified" trail in standstill condition has much  higher influence on steering moments than HTA.

Focus on the range with same sign steering and roll angles 

Regarding 7), knowing that pitch angle is a linear function of "modified" trail and a trigonometric function of HTA, the changes due to HTA are significantly damped. For example, for a given combination of roll and steering angles, a 6% variation of "modified" trail with respect to the baseline configuration will change pitch angle by the same amount while the same variation of HTA will change pitch angle by only 1.7%.

Taking into account all this, we can say that "modified" trail plays a much more important role on bicycle's stability and maneuverability than HTA. Consequently, a definition of bicycle geometry using 4 parameters ("modified" trail, chainstay length, wheelbase and BB offset) is able to fully capture all the mechanisms that have a significative role in the behaviour of the bike.

A very long post. I hope you have found it interesting. 

Trek and Specialized investigate on bicycle dynamics

After the optimisation work that the F1 chassis builder did for the carbon layup of the Venge McLaren, McLaren and Specialized continue to cooperate. This time, McLaren is focusing on studying the dynamic behaviour of bikes and analyzing how it can be improved tuning the mechanical characteristics of the frame. Bikeradar has the full story

Some interesting bits:

"The biggest issue is just how complex a bicycle is. It may seem less complex than a state-of-the-art Formula 1 car, but a bike is just a small part of the whole – the biggest factor of any bike is the rider.

The bike as a system is incredibly complex, in no small part that the ride is the integral and a highly dynamic part. Then you’ve elements like the tyre; the longitudinal and vertical deflection has an impact on performance and comfort.”

This is very well aligned with my original ideas that I have applied to develop the dynamical model. Pedal forces and reaction in the contact points have to be properly modelized to analyze dynamic behaviour of the bike. The last sentence endorse my idea that to try to capture the effect of frame stiffness on performance, it is mandatory to develop tire models complex enough.

As the McLaren engineer says, both vertical (through the relation between Crr and sinkage depth) and longitudinal deflection (through the relation between slip force and rotational stiffness of the tire) of the tire play a role on performance. I have integrated both effects in my dynamical model as you can see in the "Wheel loads" post.

"The whole research project stemmed from Specialized president Mike Sinyard’s idea that ‘smoother is faster’. It’s something the company has always thought of as true, without any real empirical factual back-up. From everything they’ve learned, Mark Cote from Specialized R&D was prepared to say: “If you can actively reduce kinetic energy losses the net gain is that you will be faster, so yes, smoother is faster. In the last six months of research we [Specialized] have learned more about bike dynamics than we have in the last 10 years.”

"It’s so early in the research partnership that no one really knows what the future will hold. McLaren could see the benefits of an intelligent bike that ‘self adapts’ – imagine a Roubaix that softens over the Pave, but sharpens up on smoother roads. McLaren hopes that it can ‘crack the logic’ of what makes a bike great. For McLaren it’s about generating the specification."

This is very interesting. I don't see a real active suspension like the one present in the Williams FW14 F1 car happening but something sleeker could be adapted to a road bike. Electroactive polymers or magnetoresistive fluids could be an option.

While all this happens, Trek is acquiring some data with an instrumented Domane for P-R.

Let's hope that some of all these interesting developments find a way to the white papers to give some scientific backup (or not) to the old mantra "Stiffer is always better".

That's all for today. Happy christmas and new year ;)

A minimal bike geometry

With this post I want to start a project that I've got in my mind for quite a long time. I've got a very busy year ahead so I don't know if I will be advancing as fast as I would like to.

As a first step, I want to define a minimal set of parameters that could determine completely the dynamic behaviour of a bike. Let's take a look to a typical bike geometry chart:

Specialized Venge 2014 geometry chart

As you can see, the number of parameters is very high and there isn't any explicit separation between parameters that affect fit (fit geometry) and parameters that affect handling (dynamic geometry). We can forget fit geometry if we consider that both stem and seatpost have infinite adjustment capability (with a Look Ergostem and a Titec El Norte seatpost for example) and, consequently, the position of the upper contact points can be set independently of the frame geometry. In short, fit is, in a strict sense, independent of frame geometry.

When talking about dynamic geometry, things are a little less obvious. Bike dynamic behaviour is affected by 3 parameters that have a relation with frame geometry: wheelbase, trail and bike+rider center of gravity position. Regarding the center of gravity, once we have defined the position of the upper contact points (stack and reach fit coordinates), the only intrinsic parameter of a frame that affects the COG position is the position of the BB with respect to the wheel axles. Consequently, a definition of bike geometry should take this into account.

The role of the wheelbase is obvious, it modifies weight balance. The last one, trail, is defined as the distance between the center of the contact patch and the intersection between the steering axis and the ground. I don't agree completely with this definition so I've defined a modified trail. In the following image you can see both definitions as a function of wheel, fork and frame parameters.

This way, the modified trail is the leverage of the steering moment generated by lateral forces in the contact patch. The following plot shows a comparison between both definitions.

The contour plot is the "traditional" trail. The surface plot is the modified one. The difference between them increases as head tube angles decrease

Taking all this into account, we can say that a minimal bike geometry from a dynamic POV can be defined with 4 parameters: wheelbase, trail, chainstay length (horizontal) and BB offset.

Next step is data gathering.

That's all for today. Thanks for reading!

Dynamical model. Wheel loads

As I've already commented, I think that slip force and rolling resistance moment are the two main loads that could play a role efficiency-wise. Consequently, I have made an effort to complexify their models as much as possible. The following 4 sheets explain the contact model and the loads acting on the wheel (chain, reactions in the axles, normal force, tangential force and rolling resistance moment).

After reading these sheets, you can imagine why the solver has problems under certain conditions. Complexity is very high.

That's all for today

Dynamical model. Modelling bike compliance

How to modelize the compliance of a bike in a dynamic model? That's a very good question. I have chosen the simplest type of model used in elastodynamics, the area that analyzes the deformation of elements in dynamic conditions. There are more complex models based in a FEA formulation of the deformable components and various choices of kinematical coordinates that can be found in some commercial codes like ADAMS Flex or Altair Hyperworks. For the moment, I didn't want to go as far.

This model is based on connecting certain elements with springs whose stiffness is derived from statical tests. It's based on linearity so the amount of deformation is proportional to the force between them. There are some features that the model isn't able to capture like the inertia associated to the deformation of the components but we will consider that it's a second order effect.

I've used this model to take into account the connection between the wheels and the frame to analyze the effect of bike compliance on performance. A simple diagram is shown below.

The wheels are free to move with respect to the frame and they are connected to the undeformed configuration of the chassis using springs. Obviously, the wheel axle and the dropouts of the bike in the deformed configuration should be coaxial so the stiffness of the springs is equivalent to the stiffness of the frame in the defined directions. Now the question that arises is: what's the stiffness of those springs and what's the relation between them and the statical tests?

Correlation between the results of static test benches and those measured in real world is a difficult issue. I recommend you to take a look at  two very interesting articles (here and here) that Damon Rinard wrote about how to improve correlation. The process that I've followed to obtain the stiffness of the vertical springs is explained in the following sheet:

As a ROT, we can say that the stiffness of these springs is half of the BB stiffness of the bike so significantly lower than the stiffness of a road tire. Some typical values are shown below:

Tour Magazin data

Once we have calculated the stiffness of the vertical springs, it's time for the horizontal ones. We can modelize the rear end as a structure clamped in the ST-Seatstay junction and in the BB and with a symmetry plane. Similarly, the fork has a symmetry plane and it's clamped in the HT. As there isn't data available under this type of loads, I've done some tests using ANSYS.

Rear end test. Steel. BEAM 188. 201 nodes. Krx=71000 N/mm for the whole rear end

A similar test was done for a steel fork with straight legs and 30mm of spacement between the crown and the dropouts. Using the same tubing than in the previous case, Kfx equals 127N/mm.

Greetings