In the last installment of the series, we discussed how the wheel diameter affects the ability to climb obstacles. If you’ve read that post, you might recall that larger wheels indeed have an upper hand when it comes to climbing over obstacles. Today we’ll investigate if they retain their advantage in smoothing out road undulations, focusing solely on the two-degrees of freedom (2DOF) model.
Please note that some of the concepts that will be used in this post were discussed at length in separate articles, to not make this one unnecessarily long and diluted. If you encounter anything you are not familiar with, you are more than welcome to explore those topics further using the links provided in the post.
Suspension transfer function
Let’s begin with the expression for the suspension transfer function in the 2DOF model. I derived it in a previous post, but let’s recall its form:
It looks quite formidable at first glance, I admit, but don’t despair. What’s most important is to note that the the shape of the transfer function is influenced by a number of parameters characterizing the suspension and the vehicle, namely:
- suspension spring rate,
- tire radial stiffness,
- suspension damping coefficient,
- tire damping coefficient,
- sprung mass,
- unsprung mass,
Thus in order to understand the benefit – or lack thereof – of using bigger wheels, we need to understand how the suspension transfer function changes with the aforementioned parameters. We can go about this in two ways. The first is to compute a lot of partial derivatives of 
and arrive at the most general, but completely incomprehensible results. The second is to select the values of the parameters corresponding to a real-world case and, varying one parameter at a time, plot and note its behavior. We will, of course, follow the latter approach.
Vehicle and parameter choice
The motivation for this series was a post on a bike forum, hence our test vehicle cannot be anything else. My choice is a Trek Procaliber 9.6 – not that I own it, but it’s easier to focus on a concrete, real-world example. Also, all parameters will correspond to how I’d set the bike up for myself.
Let’s start with the easy choices first. As per Öhlins guide, I should be using fork springs rated at 9.7 N/mm. The Trek has a 68.8-degree head tube angle, which results in an effective spring rate of 9.04 N/mm. The sprung mass is simply half the weight of the rider (me) and the bike, so 47 kg. I know, some of the bicycle mass is unsprung, but this approximation is sufficient for our purpose. Speaking of unsprung mass, I will take it to be the mass of the front wheel and half the front fork – 1.62 kg. Finally, fork damping can be easily adjusted over a wide range. Let’s start with being 979 N·s/m, corresponding to the damping ratio of 0.7. This value is a popular choice as it combines fast suspension response and small overshoot.
The choice of tire parameters is more complicated, simply because manufacturers do not list them. However, there have been some independent research papers published on the topic that might help us out. Radial stiffness has been measured by Sadauckas and Dressel for selected tire and pressure combinations, one of which is a 2.3×29″ tire pumped to 1.7 bar, giving 62 300 N/m. It happens to be exactly what I’d use in the Trek, both in terms of size and pressure. On the other hand, according to Sadauckas and Nagurka, the damping coefficient of bike tires is quite small, corresponding to a damping ratio between 0.02 and 0.05. We’ll choose the highest value, leading to being equal to 30 N·s/m.1Due to viscoelastic nature of a bicycle tire, the damping coefficient changes with speed and driving frequency. This effect is not taken into account in this post. For more information, see this article.
Suspension parameters and response
Let’s now break down the influence of each paramter on the transfer function, starting with the sprung mass, :
If you only see one line, well, so do I. The influence of the sprung mass in that range is pretty limited, even though it amounts to the rider gaining or losing a rather hefty 12 kilos.
A bigger effect can be achieved by swapping the suspension springs:
Increasing spring rate raises the resonance peak around the natural frequency at the expense of higher frequencies, which are suppressed more efficiently. In the example above, the peak value for 10.4 N/mm spring is about 1.76% higher than the reference. However, at frequency ratio equal to 10, it drops about 9.1% lower than the reference case.
A decidively more dramatic effect has manipulating the fork damping ratio:
The character of those changes is inverse to that of stiffening suspension springs: higher damping ratio decreases the magnitude of the resonance, at the same time letting more higher frequencies to pass through.
On the other hand, unsprung mass has almost no effect on the transfer function:
Yet again, the difference between the lines is hardly (if at all) discernible, even for a completely unrealistic, ultra-lightweight case of 0.1 kg front wheel assembly. Indeed, the peak values differ by less than 0.5% from reference. Again, the difference gets to a few percent for higher frequencies, but in this region the transfer function is almost zero anyway.
The exact same story can be told about the tire damping coefficient, even if we allow for unreal values:
Luckily, tire radial stiffness has a much more influence on the transfer function:
Note that the effect is inverse to increasing main spring stiffness: a stiffer tire has a less pronounced resonance peak, but lets slightly more high frequency vibrations to pass through.
Influence of the wheel
Now that we know how the suspension transfer function behaves with changing system parameters, it’s time to finally relate it to wheel diameter. The key is to establish how they are affected by the wheel size. It’s immediately obvious that swapping for a larger wheel has no direct effect on the fork spring rate or damping ratio. While the sprung and unsprung masses can shift ever so slightly, they have almost no effect on the transfer function. The same applies to tire damping.
However, smaller wheels require higher inflation pressures. According to the SRAM pressure calculator, I’d need to increase front-wheel pressure by about 13% while going from 29 to 26-inch wheels. This will affect tire radial stiffness, but it’s impossible to state how much using theoretical considerations alone, as tire properties do not depend linearly on inflation pressure.
Let’s assume an egregious difference in tire stiffnesses of 20% and see how it affects the transfer function:
As you can see, a stiffer tire actually experiences a smaller resonance peak. The difference, however, is rather small and amounts to just 2.7% at the peak. It grows somewhat for higher frequencies, but in this region, the transfer function is so close to zero anyway, that it hardly makes any difference. Furthermore, the real-world difference in radial stiffness between a 26″ and 29″ wheel is likely much smaller than what we assumed.
Interestingly, the effect a stiffer tire has on the transfer function can be offset by using stiffer springs and lower damping. For instance, by stiffening the suspension by 10% and decreasing damping by about 1.5%, 26″ wheels come exceptionally close to the Trek:
Is it worth it?
So, does using bigger wheels offer a significant advantage when it comes to suspension function? Well, that depends on your use case. For an everyday Joe, I’d say it makes no difference. But if you’re racing in cross-country or downhill races, you might have a marginal – really marginal – benefit in the frequency ratio range between about 1 and 10. For the Trek, it would correspond to wavelengths between 140 and 14 cm ridden at 30 km/h. In real-world terms, it would be equal to putting the hammer down on a corrugated road. Other than that, I doubt you’ll notice anything. But hey, it’s all about marginal gains, right?
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