Bike Think: Bicycle frame stiffness and pedaling efficiency

Frame Flex

What effect does frame stiffness have on efficiency?

When you push on the pedals, the frame deflects like a spring. This deflection absorbs energy known as strain energy. It has long been assumed that all or most of this energy is lost. The first law of thermodynamics states that energy cannot be created or destroyed. So "lost" means that the strain energy stored and then released by the frame goes somewhere besides the drive train. So where does this energy go? Is it really lost or can the frame push back in such a way that the energy is routed into the drive train?

What is work energy?

First, it is critical to get a firm understanding of what work energy is. Work energy is defined as a force times the movement in the direction of the force and at the point of the applied force (W=Fd). A cyclist going down a hill looses potential energy proportional to the elevation change, not the distance traveled along the slope. The work done on the cyclist by gravity was only in the direction of the gravitational force (W=Fd).

Consider a mass sitting on top of a rigid block. The mass has gravitational potential energy based on the elevation of the mass. The mass is exerting a force on the block. But the mass is not doing any work because there is no movement.
Now lets say that the rigid block was replaced by a compression spring. The mass will move down some and compress the spring. The mass lost some gravitational potential energy from the distance it moved down. The work done on the spring by the mass was the gravitational force on the mass times the distance moved (W=Fd). Once the mass moves down to the point of equilibrium and stops, no more work is being done. A quantity of gravitational potential energy of the mass has been transferred to strain potential energy in the spring. The mass continues to exert a force on the spring, but there is no movement. There is no energy being transferred to the spring. No work is being done.
Suppose you replace the mass with your foot. When you push the spring down to the same compressed position, you do the same amount of work on the spring as the mass did. Now you hold the spring there. Hold it there for 24 hours. You say, "I'll get tired. Of course I'm doing work." The effort that you are exerting is just the inefficiency in the way your muscles maintain a force. But that's all internal to your body. The spring sees no difference between the force from the mass and the force from your foot. There is no energy being transferred to the spring. No work is being done to the spring.

Can applied forces be separated?

The other key concept is the structural analysis principle of superposition. This states that if one or more forces are applied to a structure, the deflections from each force can be found independently and then added together. It also means that you can analyze the reactions of a structure independently, regardless of what other loads may be applied to the structure. This is critical for us, because a bicycle frame has many different and changing loads applied to it. We need to be able to isolate one force at a time and look at what the frame's reaction is to it. So we can examine what happens to the frame under pedaling forces without worrying about what forces are being applied at the handlebars, seat, or dropouts.

Bottom Bracket Reaction Forces

So we're finally ready to start talking about what happens to that mysterious strain energy. The principle of superposition comes in handy now because we can separate out the pedaling reaction loads on the bottom bracket. I have divided them up into these four loads applied at the center of the bottom bracket:

	Horizontal Force F_H: The positive direction for this force is pointing from the bottom bracket toward the front wheel.
	Vertical Force F_V: The positive direction for this force is pointing up from the bottom bracket.
	Vertical Torque T_V: This is a torque around the vertical axis. It is caused by horizontal reactions at the pedals. This is a significant contributor to frame strain energy for left pedal forces only. The right pedal forces are close to in line with the chain, so the total torque on that side is small.
	Horizontal Torque T_H: This is a torque around the axis which points from the bottom bracket toward the front wheel. It is caused by vertical reactions at the pedals. This is the largest contributor to frame strain energy. It is also the most difficult to understand how the strain energy might get back into the drive train and not be lost. So we'll isolate this one and look at it in detail.

Any bottom bracket loading condition from pedaling forces can be represented by a combination of these four loads. So we can analyze the frames response to each of these loads separately and add up the responses later.

Frame Strain Energy

To find how much work energy is applied at the pedal, we only need to know the force and how far the pedal moved in the vertical direction. When the cranks are horizontal and you apply a vertical force at the cranks, the pedal moves down causing the cranks to rotate, which causes the chain to move. The movement in the chain times the reaction tension in the chain is the work energy delivered to the rear wheel. But there will be a small additional vertical pedal movement. The pedal force causes reaction forces at the chain and bottom bracket bearings. The vertical downward pedal force causes reaction forces that push down on the closer bearing and up on the opposite bearing. These two forces create a torque reaction at the bottom bracket. The frame's lateral flexibility allows the bottom bracket to swing in reaction to this torque. The closer bearing will go down and the opposite bearing will go up. In the meantime, the whole bottom bracket will move laterally. In fact the lateral movement will be larger than the vertical movements of the bearings. However, the work energy done at the bottom bracket to the frame is the force at each bearing times the motion in the direction of those forces. This work energy gets temporarily stored as strain energy in the frame. So the vertical travel of each bearing is what dictates how much energy was put into the frame, even though the lateral deflection is more noticeable. This is similar to a cyclist going down a hill. The work done on the cyclist by gravity is only related to the vertical distance that the cyclist travels, even though the horizontal distance down the hill will be much greater.

Simplified Example

It is somewhat complicated to analyze the forces in the crank and bottom bracket system. All the sine's and cosine's get in our way of seeing what is happening. So let's take a look at a simpler example first.

Mount a drive side crank and bottom bracket on top of a compression spring. Put the bottom bracket on a frictionless cylinder so it can only move up and down with the spring deflection. With the crank just past the top, apply a vertical force to the pedal. Gradually increase the force until the crank is horizontal. Continue applying the force but gradually decrease until the force is zero at the bottom.

The fan graph below illustrates what would happen as you apply the force. Each line on the graph represents the position of the crank in 11.25° rotation increments. Each increment also represents a 20.6mm movement of the chain. The vertical force is varying from 0 at the top, to 500N (or 112 pounds) at horizontal, to 0 at the bottom again.

As the force increases from 0° to 90°, the bottom bracket moves down as the spring is compressed. As the spring compresses it absorbs energy. As the spring deflects, the pedal moves down more than it would if it were mounted to a rigid frame during this period. This difference in movement times the pedal force is the energy that goes into the spring

From 90° to 180°, the force decreases and the spring releases energy as it pushes the bottom bracket back up. The spring is releases the energy that was stored in the first part of the pedal down stroke. The crank rotation continues as the bottom bracket moves up and the pedal vertical movement slows. The spring is returning the strain energy by pushing the bottom bracket up and thereby contributing to the crank rotation. This contribution to the crank rotation directly contributes to the chain movement, and therefore directly transfers work energy to the chain. When the pedal vertical position slows and stalls near the bottom of the stroke, the crank is still rotating and the chain is still moving at a constant rate. The energy to rotate the crank in the bottom of the stroke is provided by the spring. So 100% of the strain energy in the spring gets returned to the drive train. Keep in mind that this vertical deflection is exaggerated compared to a bicycle frame in order to help visualize the deflection. Click here for an animation showing what happens to the spring strain energy in this example.

Real World Example

Try this experiment. It will work best if you mount your bike to a trainer and disengage the resistance roller.

Put the cranks in the horizontal position.
Place a rigid block or stool under the forward pedal so that there is a small gap under the pedal.
While holding the rear brake firmly, stand on the pedal so that it is pushed down to the stool.
Keep holding the pedal down and release the brake.

When you pushed the pedal down, the chain did not move since the brake locked the rear wheel. Since the chain did not move, no work energy was delivered through the chain. The crank moved down with the pedal as the frame was strained. When you released the brake, the frame was able to move the center of the crank back up to relieve the strain energy. But the pedal remained in it's lower position, so the crank had to rotate around the pedal as the bottom bracket went up. There was a reaction force in the chain and the chain moved as the crank rotated around the stationary pedal. The strain energy of the frame was converted to rotation kinetic energy in the wheel.

FEA Model Quantifying Strain Energy

For a final part of this analysis, I have used an FEA model to determine the response of a frame to the four loads described earlier. It is somewhat trivial to look at how much energy goes in and out of the frame since we know that it gets released into the drive train. But it is interesting to understand about how much energy gets temporarily stored in the frame.

The FEA model is a 54cm True Temper RC2 chromoly steel frame. See my FEA model correlation section to see how this model compares to several real frames. It is certainly accurate enough to examine the general response of a frame to pedaling loads. For a more detailed description of a beam FEA model such as this, see "Finite-Element Structural Analysis: A New Tool for Bicycle Frame Design" by Leisha A. Peterson and Kelly J. Londry. It is a good description of the application of FEA to a bicycle frame, although they start out with the assumption that frame strain energy is lost.

I used the FEA model to determine how much energy per force unit the frame would store for each of the four loads. You can see the FEA reaction to each of these loads by clicking on one each of the following links:
Horizontal Force F_H
Vertical Force F_V
Vertical Torque T_V
Horizontal Torque T_H
I then used published pedaling force curves to find the reaction forces at the bottom bracket and chain throughout the pedal cycle. These reaction forces were then multiplied by the energy constants from the FEA to determine how much strain energy is in the frame at every point in the pedal cycle. I also created a graph to compare the energy rate, or power, of this strain energy to the total power output of the cyclist.

Conclusion

Having concluded that frame flex does not waste energy, I do not believe that frame stiffness is irrelevant. You could say that a stiff frame feels more responsive. A stiffer frame can give the rider more confidence especially in a sprint. I think the fact that you don't have a "stiffer is always better" criteria makes frame design that much more interesting.

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