I have a Specialized Roubaix with a 7r carbon frame; the Tarmac has 11r. Will I notice a speed change if I ride "exactly" under the same conditions? Does the additional 4r = more speed? I am thinking in upgrading but I dont want to make the investment without some good feedback.
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In real terms unless you are very top level cyclist it will make very little difference. There's an awful lot of focus on bike in the road cycling world, when in reality its the person that makes the difference to how fast you are. Ride what you are most comfortable on.
The notion that stiffness equals greater performance is more true in the lab than on the road. Sean Kelly won hundreds of races - often in sprints - throughout the 80s and he won most of those on a Vitus 979 aluminum bike that was probably the most flexible bike used in professional racing in the past 50 years.
It's pretty unlikely you'll notice much difference between the two frames. Also keep in mind that for the comparison to be accurate every single other part on the bike would have to be identical. Bars, stems, wheels, cranks all contribute to stiffness.
As with motoring, you can use stiffer engine mounts so as to help transfer more power from the engine to the wheels. However, I'm told that it can make for a hideous ride quality in the vehicle. Now Imagine that on an ultra stiff road bike. Great on silky, smooth, new tarmac - but in the real world, wincing on every little bump in the road will affect your speed negatively. I know I stop pedalling and wince as I roll over rough bits of road. Mind you- great for powering up the hills I expect.
General rule of thumb, a stiffer frame will absorb less of the input energy and transfer more energy - hence more power from you legs means more power to the wheel.
The bikes have different designs, so the aerodynamics of the bikes and the rider on them will be different which will result in different speeds.
A stiff frame will transmit the road imperfections into the rider causing fatigue and causing the rider to expend more energy because the bike/rider is being lifted up by road imperfections where as the more flexible bike will absorb them meaning the bulk of the mass is moving forwards.
An ideal bike is stiff where it needs to transmit peddling force, but has shock absorbent qualities to absorb bumps and road imperfections while being aerodynamic for both bike and rider.
The stiffer a frame is, the more it resists flexing under load.
For the sake of demonstration, say you have a carbon frame that under a load of 400 watts of pedaling power has 2" of flexion. The actual power going to the wheels is 400 - Cost to flex.
Now you take a frame that only flexes 1". Less of your power goes into flexing the frame, so more power gets translated to the wheels, which results in a faster bike.
The stiffer a frame, the harsher the ride in general, so it may not be quite as comfortable.
If you are familiar with cars at all, it's the same concept behind horsepower at the crank, versus horsepower at the wheels.