Saturday, February 8, 2014

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 angle
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'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. 

Wednesday, December 25, 2013

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 ;)

Friday, November 29, 2013

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!

Saturday, September 21, 2013

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

Thursday, September 19, 2013

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 componentand 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 ishown 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

Wednesday, September 18, 2013

Thoughts about chassis stiffness and efficiency

As an introduction to the upcoming post about how I modelized frame compliance in the dynamical model, I would like to give a general overview about some possible links between frame compliance and efficiency.  

First of all, we shouldn't forget that the bike is controlled by a rider whose input could be modified by the mechanical properties of the frame and the components. For example, a rear end too stiff could cause excessive bouncing of the rider and have a negative effect on his power output. For this reason, we should always consider the negative effects that could have a particular design feature on the vibration in the contact points, intersegmental loads, joint torques and steering inputs and not treat the bike as an isolated system.

Leaving this influence on the rider input aside, I have identified some possible efficiency loss mechanisms due to the deformation of the chassis:

- Rolling resistance. Although the rolling resistance coefficient (Crr) has been always defined as a constant value, there is a relation between it and sinkage depth. The explanation for this is pretty simple: Rolling resistance moment is caused by the difference in normal pressure  between the leading and the trailing edge of the contact patch due to the hysteretic cycle of the material. If sinkage depth is increased, the tire deforms more, the severity of the hysteretic cycle is maximized and the losses increase. There is also second order effects like the radius of the contact point between the ground and the tire.

- Drivetrain misalignement. Both the torques perpendicular to the BB axle caused by the pedalling forces and the combination of asymmetric chain loads and symmetric rear ends produce misalignements between the BB axle and the rear wheel axle. Those misalignements could cause torsional loads on the chain and increase friction due to the contact between the side plates and the sprockets/chainrings.

- Sideslip and camber of the rear wheel. Once again, the combination of asymmetric chain loads (both in the longitudinal and horizontal planes) and symmetric rear ends produce two effects: 1) a misalignement between the bike speed and the speed of the contact point with respect to the bike and 2) a small camber angle. Dissipation increases due to the presence of a yawing moment that tends to align those two speeds and the effect of camber on rolling resistance.

- Wheel slip. Traction is a function of wheel pressure and frame/fork stiffness so an optimization of these parameters could minimize slip and, consequently, power losses.

- Losses in the frame. The harmonic excitation of the bike causes losses in the structure due to both hysteresis and viscoelaticity that can affect negatively the performance of the bike. Alternatively, a well tuned placement of materials with these characteristics can increase damping and, consequently, comfort.

As you can imagine, the analysis of such complex interactions would need a very complete system. A deformable 3D bike model controlled by a virtual rider capable of balancing the bike in a similar way an human would do would be needed. Additionally, both the effect of drivetrain misalignement and losses in the frame have to be quantified using experimental methods or complex FEA models.

In my case, I haven't gone so far. I have modelled just two of these mechanisms: rolling resistance and wheel slip. I think that these are the ones that could play a major role on efficiency.

That's all for today. Greetings

Thursday, September 12, 2013

Dynamical model. Introduction

Those of you that follow this blog from some time ago know that the development of an accurate bike model was one of my main objectives. From the first model that I presented here (very similar to the one that is used by analyticcycling.com), passing by a second one that was richer, I have tried to increase complexity progressively to give answer to some questions that, to the best of my knowledge, nobody has answered. Those questions appear regularly in bike forums without a clear answer, showing that there is still job to be done to explain properly the link between the traditional methods for quantifying bike performance and how a bike behaves in real world. Bike design and construction have improved enormously in the last few years and I think that the effort done to explain scientifically how those improvements are beneficial to the consumer isn't enough.

After those two simpler models, I decided to devote more time to the development of the dynamical model. It started with a tire model able to handle discontinuous contact. Later, I tried to build the whole system around this tire model without success so I decided to step back and add elements to the model progressively.

A virtual ergometer was the first sub-model built. Everything worked flawlessly until I decided to include the elastic couplings between the torso and the bike (arms and saddle) so that feature was removed. Next, the bike-wheels sub-model was built with better results so I only needed to merge those two models. Once again there were problems so I have to calculate saddle loads using the virtual ergometer and use those forces as input in the full model.

I have to say that this trial and error procedure is really frustrating. When you spend some days/weeks gathering all the data needed to add a certain element to the model and integrating it and finally the software fails to solve and it doesn't give a meaningful explanation to the error, you want to give it up. I could have used other solving procedure and maybe this would have speed up the process but I also value all the things that I have learned finding the limits of the solver. For example, a dynamics program (Adams, Working Model or SIMPACK) could have done the job but I doubt I could have integrated the most complex and interesting features.

It's isn't perfect yet but I think that it is accurate enough to give some interesting answers for the future direction of bike design.

A quick animation. Take a look at how the BB and the chain moves during the downstroke ;)


More soon. Thanks for reading!