Ride Feel

Ride character is primarily affected by three factors: frame stiffness, natural frequency response, and histeretic damping.

Frame Stiffness

figure 1

The impacts that reach your body are related to the stiffness of the frame in the direction of the applied shock loads. This quality is primarily noticeable with larger dynamic loads such as potholes or larger bumps. The frame stiffness is a combination of frame/tubing geometry and material stiffness. Figure 1 shows the relative stiffness for seat stays for common seat stay tubing in each material.  The results are from my beam FEA frame model.  What is interesting is that the aluminum seat is the least stiff and the carbon fiber is the stiffest.  Of course the stiffness in each case can be changed by changing the diameter or thickness of the tubing, or by putting a bend in the seat stays.  But I think this comparison of common seat stays in each material suggests that frame stiffness is not the primary driver in the ride feel of a frame.  The deflections for the loads applied in the FEA model were around 0.5mm for a 220 pound force.  A seat stay assemble that flexes 0.34mm is almost twice as stiff as one that flexes 0.65mm.  But they are both going to deliver most of a shock load to the rider.

The impacts that reach your body are related to the stiffness of the frame in the direction of the applied shock loads. This quality is primarily noticeable with larger dynamic loads such as potholes or larger bumps. The frame stiffness is a combination of frame/tubing geometry and material stiffness. Figure 2 shows the relative stiffness for seat stays for common seat stay tubing in each material.  The results are from my beam FEA frame model.  What is interesting is that the aluminum seat is the least stiff and the carbon fiber is the stiffest.  Of course the stiffness in each case can be changed by changing the diameter or thickness of the tubing, or by putting a bend in the seat stays.  But I think this comparison of common seat stays in each material suggests that frame stiffness is not the primary driver in the ride feel of a frame.  The deflections for the loads applied in the FEA model were around 0.5mm for a 220 pound force.  A seat stay assemble that flexes 0.34mm is almost twice as stiff as one that flexes 0.65mm.  But they are both going to deliver most of a shock load to the rider.

Natural Frequency Response

figure 2

Natural frequencies can exaggerate rough road and other vibrations. In general, a higher natural frequency is desirable because it reduces the chance of a vibration in the frame that matches a natural frequency. The natural frequency of the frame is affected by a multitude of factors including frame geometry, rider’s position, and how full the water bottle is.

figure 3

In general, frames with higher stiffness to weight ratio’s will have higher natural frequency responses.  So materials with higher stiffness to density ratio’s will tend to produce frames with higher natural frequencies.  But stiffness and mass are also influenced by tubing geometry (diameter, thickness, etc.).   Figure 2 shows relative stiffness to density ratios.  Figure 3 compares first mode natural frequency response from the same FEA model as above.  Figure 3 implies that aluminum is less harsh than steel or titanium.  Obviously there is more to the picture.

Frame Material Histeretic Damping

Figure 4

figure 5

Smaller vibrations from riding on non-smooth surfaces carry through the frame to your body.  These vibrations are diminished or eliminated by the damping, or histeretic, effect of the frame material.  This effect would be noticed as the frames smoothness or liveliness. Also, material damping reduces the effect of any natural frequencies.

The illustration below shows a simple model of what is going on in the frame.  The red coil is a perfectly elastic spring.  Next to it is a damper.  If the mass is pulled up and let go it will start to oscillate or vibrate up and down.  Without the damper (and in a vacuum), it would continue to move up and down forever.  The energy continually cycles between kinetic energy in the mass to strain energy in the spring.  But the damper dissipates the some of the energy each cycle, converting it into heat.  The “damper” in frame materials is called the histeretic effect.  It can be thought of as the material’s internal friction.  In materials used in bicycle frames this damping effect dissipates about 1% of the strain energy each cycle.  The histeretic damping causes vibrations in the frame to decay.  Materials such as aluminum with a low damping ratio will resonate longer after an impact.  The histeretic damping effect also muffles natural frequency responses.

figure 6

The response spectrum graph (figure 6) shows how each material would respond to input vibration across a range of frequencies. I excluded titanium due to a lack of reliable damping ratio data.  In this plot, carbon fiber exhibits a higher and muffled response.  Aluminum shows to be very excitable.  This plot illustrates well the meaning of the expression “steel is real”.  It is more responsive than carbon fiber.  But not as responsive, or some would say harsh, as aluminum.

The three graphs below are representations of the first 12 modes of natural frequency superimposed on each other.  They give an indication of what the frame’s ride feel is like.

Steel

Aluminum

Carbon Fiber

Thanks to the following for contributions to this article:

Crisp Titanium,    Calfee Design,    Don Walker Cycles

Chainstay Bridges

This is a brief look at the stress and stiffness effects of chainstay bridges.  I applied the horizontal torque load from pedaling to my beam FEA model with and without a chainstay to compare stiffness.  The FEA shows about a 5% increase in stiffness with a chainstay.

However, there appears to be no significant change in stress levels.  Real stress levels would be sensitive to local stress concentrations at all joints and corners, so it is difficult to definitively say that there is no difference.  The FEA stress plots below give a general sense of the stress patterns under pedaling loads.

Joint Stress

The lugged frame has found itself relegated to a dedicated niche market.  For metal frames, lower cost, infinite geometry choices, and ability to weld aluminum and titanium has made welded frames dominate.  Whatever your preference may be, the comparison of the stresses at the joints for these two joint methods provides some interesting insight into joint design of a bicycle frame.

It is important to remember the difference between strength and stiffness. The strength of a frame determines if it will break when you hit a pothole.  The stiffness of a frame determines how much it bends when you pedal and how much it hurts when you hit the pothole.  Stress patterns are only dependent on loading conditions and geometry (including angles, tube diameter, wall thickness, cross sectional shapes, etc.)  For a given stress pattern, the material properties determine strain and deflection, or how much the frame will flex.  Because the local stresses at the joints cover such a small portion of the overall frame, they are not a significant factor in overall frame stiffness.  These local stresses are only important in the frames strength and durability.

Below is an FEA comparison of a lugged and welded down tube to head tube joint.  The load is a torsion applied to the down tube with the ends of the head tube fixed.  The arbitrary torsion load is the same in each case.  The plotted stress colors are on a consistent scale.

The maximum stress in the welded joint is about double that of the lugged joint.  But keep in mind this is a rough comparison.  Factors such as lug design and weld fillet thickness will significantly affect the stress pattern in each case.

The lugged joint has two advantages as far as stress is concerned.  First, it has added material in the stress concentration area.  The lug material reduces the stress in the joint.  Secondly, the edge of the lug is shaped and tapered to avoid a concentrated joint edge where stress would concentrate.  Although, you can see a definite step in stress at the edge of the lug on the down tube.

This is not to say that welded frames are bad.  Obviously it is possible to build welded frames that are strong enough.  However, the lugged frame may be able to use thinner wall tubing if it were stiff enough.  Looking at these FEA plots, you can see why many carbon fiber frames flare out at the joints to avoid stress concentrations.

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 loses 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:

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.

The animation (Crank_on_spring_energy) shows what happens to the spring strain energy in this example.  This animation shows what happens to the energy as it is stored and released from the spring that the crank is mounted on.  As the spring is compressed, the energy delivered to the chain lags behind the energy delivered to the pedal.  When the energy is released from the spring and the bottom bracket is pushed back up, the chain energy catches back up to the pedal energy.   Notice how the bottom bracket center point moves down and up on the crank arm position graph.

Real World Example You Can Try

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

1. Put the cranks in the horizontal position.
2. Place a rigid block or stool under the forward pedal so that there is a small gap under the pedal.
3. While holding the rear brake firmly, stand on the pedal so that it is pushed down to the stool.
4. 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.

FEA Correlation to real tests

The FEA model is a 54cm True Temper RC2 chromoly steel frame.  I subjected the FEA model to the load cases from the Rinard Frame Stiffness Test.  These results were compared to test results from 3 different 54cm steel frames.  The results were very close and definitely good enough for studying the general behavior of frames under pedaling load.

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.

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.

See also:

• Off the Beaten Path:  Science and Bicycles:  Frame Stiffness
• Frame Flex by David Kirk