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Willem Den Boer

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Test and modeling results are reported on a bicycle crankset with limited elasticity. Like record-breaking running shoes, the crank set has spring action which mitigates the effect of the dead zone during the pedal stroke. Fiber composite leaf springs are inserted inside the hollow carbon crank arms. The crank arms are not directly attached to the crank axle. Instead, sleeve bearings allow the crank arms to rotate by up to about five degrees relative to the crank axle. The rotation is counteracted by the springs and is proportional to applied torque at the pedals. The novel crank set and a conventional crank set with forged aluminum crank arms were both tested on a stationary bike. The ratio of effective speed to input power is used as a measure of cycling efficiency. Depending on the difference in torque during the downstroke and in the dead zone, this ratio is typically a few percent higher for the novel crankset than for a conventional crankset. Multiple tests show efficiency improvements in the range of 1 to 4% at power levels of 200 W and cadence of 71 rpm with average of around 2%. Details of a test with 2.3 % improvement are presented. This would translate, for example, into a one minute advantage in a 45 minute time trial. In an attempt to understand the test results computer modeling of bicycle speed and crank arm angular velocity vs. time was performed for non-elastic and and elastic crank arms. It is difficult to explain the test results with computer modeling unless it is assumed that conventional crank sets introduce energy losses in the drivetrain from twisting of the crank arms and flexing of the bicycle frame under load at the pedals and that these energy losses are reduced for the crank set with built-in elasticity.

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Type of the Paper: Conference Paper

Improvement of Cycling Efficiency for Drivetrains with Elasticity

Willem den Boer

Huron Cycling LLC, Brighton, Michigan, USA; willem@huroncycling.com, ORCID 0000-0001-9290-8320

Name of Editor: Jason Moore

Submitted: 01/09/2023

Accepted: 07/09/2023

Published: 07/09/2023

Citation: Den Boer, W. (2023). Improvement of Cycling Efficiency for Drivetrains with Elasticity. The Evolving Scholar - BMD 2023, 5th Edition.

This work is licensed under a Creative Commons Attribution License (CC-BY).

**Abstr**act:

Test and modeling results are reported on a bicycle crankset with limited elasticity. Like record-breaking running shoes, the crank set has spring action which mitigates the effect of the dead zone during the pedal stroke. Fiber composite leaf springs are inserted inside the hollow carbon crank arms. The crank arms are not directly attached to the crank axle. Instead, sleeve bearings allow the crank arms to rotate by up to about five degrees relative to the crank axle. The rotation is counteracted by the springs and is proportional to applied torque at the pedals. The novel crank set and a conventional crank set with forged aluminum crank arms were both tested on a stationary bike. The ratio of effective speed to input power is used as a measure of cycling efficiency. Depending on the difference in torque during the downstroke and in the dead zone, this ratio is typically a few percent higher for the novel crankset than for a conventional crankset. Multiple tests show efficiency improvements in the range of 1 to 4% at power levels of 200 W and cadence of 71 rpm with average of around 2%. Details of a test with 2.3 % improvement are presented. This would translate, for example, into a one minute advantage in a 45 minute time trial.

In an attempt to understand the test results computer modeling of bicycle speed and crank arm angular velocity vs. time was performed for non-elastic and and elastic crank arms. It is difficult to explain the test results with computer modeling unless it is assumed that conventional crank sets introduce energy losses in the drivetrain from twisting of the crank arms and flexing of the bicycle frame under load at the pedals and that these energy losses are reduced for the crank set with built-in elasticity.

**Keywords:** Cycling Efficiency, Bicycle Drivetrain,
Bicycle Crankset

It is well known that carbon fiber plates in running shoes provide a propulsive sensation which helps improve speed. Several brands of shoes have been using this technology to add a degree of elasticity in their state-of-the-art products. This has led to multiple world records in road and track running in recent years.

In the drivetrain of bicycles a limited degree of elasticity can be
built in as well with beneficial effects, for example in the spider of
the crank set (Hamamoto, 2019) or in the crank arms (den Boer, 2019-1
and Bastianelli *et al*, 2019). In figure 1 the method we adopted
is shown. Fiber composite leaf springs are inserted inside the hollow
carbon crank arms. The crank arms are not directly attached to the crank
axle. Instead, sleeve bearings allow the crank arms to rotate by up to
about five degrees relative to the crank axle. The rotation is
counteracted by the leaf springs and is proportional to applied torque
at the pedals. Since torque varies from a maximum during the downstroke
to a minimum in the dead zone of the pedal stroke, the leaf springs
inside the hollow crank arms bend and store energy during the downstroke
and release this energy in the dead zone in the direction of rotation to
contribute to effective torque.

**Figure 1**. Cutaway view of crank arm and crank
axle.

The angle of rotation *θ* of the crank arms relative to the
crank axle is given by:

\(\theta = \frac{\tau}{k} = \frac{P}{\omega k}\) (1)

where *τ * is the torque from the tangential force
*F _{p}* applied at the pedals (

Figure 2 shows a photograph of several crank sets based on the
principle of internal leaf springs. In earlier prototypes we also used
metal compression springs outside the crank arms (Bastianelli *et
al*, 2019, www.huroncycling.com) for proof of concept. They are,
however, too heavy for practical use.

**Figure 2**. Photograph of crank sets based on the
design of figure 1.

Although the principle of operation of limited elasticity in bicycle drivetrains is quite different from that in running shoes, testing shows a benefit in cycling as well.

**Testing**

The novel crankset was installed on a carbon frame bicycle and compared with a conventional crank set with forged aluminum crank arms, while keeping the gear ratio constant at 36/17 and the slope constant at 3 %. To eliminate wind drag factors and changing conditions the testing was performed indoors on a Tacx Neo smart trainer with two power meters (see figure 3):

A Powertap P1 pedal power meter at left and right pedals to measure input power

The internal power meter of the Tacx Neo to measure effective speed

**Figure 3**.Test setup

Data from the smart trainer was displayed and stored on the Tacx app on a smart phone via Bluetooth wireless protocol.

The input power was measured at the pedals, both left and right, with a Powertap P1 pedal-based power meter. This power is representative of the input power applied by the left and right legs to the pedals. The information was displayed and stored on a Garmin Edge 810 head unit via ANT+ wireless protocol.

More than twenty sets of tests were performed of 15 or 30 minutes duration each with seven different prototypes of our novel crank set. To verify repeatability of the measurements each set of experiments consisted of at least two tests with the novel crankset with limited elasticity and at least two tests with the conventional crank set made of forged aluminum. All other conditions in a set of tests, such as speed target, slope setting, gear ratio) were kept identical. This allows a direct comparison of the two crank sets. Each test was performed at constant input power of around 200 W.

The speed-power ratio *SPR* was used as a measure for cycling
efficiency. Percentage of change *∆(%)* between the prototype
novel crank set and the conventional crank set was defined as:

\(\mathrm{\Delta}(\%) = \frac{{SPR}_{proto} - {SPR}_{conv}}{{SPR}_{conv}}\) (2)

where *SPR _{proto }*and

Table 1 shows the input conditions for a representative test of 15 minutes. The effective speed measured on the Tacx smart trainer is based on the number of rotations of the back wheel.

**Table 1.** Experimental setup and conditions of
typical test

Test equipment | Tacx Neo smart bike trainer |
---|---|

Slope setting | 3 % |

Power target | 200 W |

Duration of tests | 15 minutes |

Gear ratio | 36/17 |

Power meters | Tacx Neo (better than 0.5 % accuracy) |

Powertap P1 left and right pedal power meter | |

Control conventional crank set | FSA Gossamer BB30 175 mm crank length |

Prototype novel crank set | Leaf springs with k=1000 Nm 175 mm crank length |

The conventional crank set was an FSA Gossamer BB30 unit with forged aluminum crank arms. After the control testing, the conventional crank set was replaced with the novel crank set with embedded leaf springs. The slope on the Tacx Neo was set at 3 % to prevent any freewheeling during the tests. In each test the power was ramped up to 200 W before starting the measurement and was then maintained around 200 W for the remaining time of the test.

Test results for a typical 15 minute test are shown in Table 2. Repeatability of results was within 0.5 % and the table shows the average of two identical tests for each crank set.

**Table 2.** Test results around 200 Watts on Tacx Neo
at 3 % slope for 15 minutes

Conventional crank set | Novel crank set | |
---|---|---|

Average Speed Tacx (km/h) | 19.4 | 19.4 |

Average Cadence Tacx (rpm) | 71 | 71 |

Distance Tacx (km) | 4.865 | 4.865 |

Average Power Powertap P1 (Watts) | 200.5 | 196 |

Energy Powertap P1 (kJoule) | 180.5 | 176.5 |

As seen in Table 2, for the novel crank set the speed-power ratio is
higher than for the conventional crank set. The difference
*∆(%)*, as defined in equation (2) is 2.3 %. In other words: With
the novel crankset the cyclist can go faster using the same power or
maintain the same speed while exerting less power. This is confirmed by
the energy in kJoule comparison, which gives the same percentage
difference. It takes less energy to complete the same distance at the
same speed for the novel drivetrain.

The twenty sets of experimental data for seven different prototypes
with *k* = 1000 Nm show a range of improvement *∆(%)* of 1
to 4% as compared to the conventional crankset with forged aluminum
crank arms.

The lower power needed to maintain the same speed with the novel crank set may be explained by the reduced force on the pedals during the down stroke at 3 o’clock and 9 o’clock crank angles and the release of stored energy from the springs in the dead zone.

A few tests were also performed at constant 200 W power for different cadence levels by changing the gear ratio to 36/19 and 36/15. In the range of 64 to 80 rpm the improvement with the novel drivetrain remains in the same range.

For constant cadence of about 72 rpm but lower power levels of 179 Watts (gear ratio 56/19) we still see similar improvement. When the power is further lowered to 162 Watts (gear ratio 56/21) at 72 rpm on the Tacx trainer, the improvement is reduced. This is to be expected as a result of the reduced force on the pedals and reduced torque on the crank and therefore reduced deflection at the pedals. The test results do not depend on the type of pedal-based power meter: For earlier testing a Garmin Vector power meter for left and right pedals was used with similar results as the Powertap P1.

**Modeling**

In an attempt to understand the effect of drivetrain elasticity on
bicycle speed vs. input power, computer modeling was performed. Figure 4
shows the forces at work and the resulting speed of the bicycle. Torque
*τ *on the pedals *τ *= *F _{p}*r*, where

**Figure 4**. Forces at work on a bicycle

Measurements of torque vs crank angle by us (den Boer, 2019-2) and
many power meter manufacturers show that the combined power *P*
applied at left and right pedals is approximately sinusoidal with a
frequency twice the crank rotation frequency. In the following
*P* is assumed to vary between a maximum during the downstroke
and zero at the dead spot:

\(P = P_{0}{{(sin}{\omega t})}^{2} = P_{0}(1 - \cos{(2\omega t))/2}\) (3)

where *ωτ *is the crank angle and *ω* is the angular
velocity of the crank arms. *P _{0}* is the maximum power
during the downstroke when

\(\tau = \frac{P}{\omega}\ \) (4)

The force propelling the bicycle is given by:

\(F_{bike} = \frac{P}{v}\) (5)

The change in speed *dv* of the bicycle from a starting
velocity of *v _{0}* is given by Newton’s law:

\(dv = \frac{F_{net}*dt}{m}\) (6)

where *F _{net}* is the net force on the bicycle
(

In the following the counterforce is gravity on an uphill slope with
angle *α *:

\(F_{counter} = mgsin\alpha\) (7)

where *g* is the gravitational acceleration (*g* = 9.81
m/s^{2})

**Modeling of conventional drivetrain without
elasticity**

For a drivetrain without any elasticity the angular velocity of the crank arms and velocity of the bicycle are directly related to each other:

\(v = \omega*R*GR\) (8)

where *R* is the rear wheel diameter and *GR* is the
gear ratio between front chainring and back cog. The following is an
example calculation for the following input parameters:

*m* = 70 [kg]

*v _{0}* = 1 [m/s]

*P _{0}* = 500 [W] for an average power of 250 W

*R* = 0.35 [m]

*GR* = 2

*r* = 0.175 m

Uphill slope *α* = 0.1 rad, corresponding to 5.72 degrees or
6.37 %

**Figure 5**. Example of calculated
*F _{bike}, F_{counter}* (dashed line) and
velocity and angular velocity (dotted line) vs. time for a drivetrain
without elasticity. LEFT and RIGHT represent alternatingly left and
right pedal.

In figure 5 the results are shown for the applied force
*F _{bike}* to the bike and the resulting velocity of the
bike and angular velocity of the cranks vs. time.

\(\mathrm{\Delta}v\ (\%) = \frac{v_{\max} - v_{\min}}{v_{\max} + v_{\min}}\) (9)

**Figure 6**. Dependence of velocity oscillations on
cadence

It is well-known that the most energy-efficient way to travel is at constant velocity. If the velocity oscillates, the power required to maintain the average final speed of 3.65 m/s (the pace) increases, as a result of the quadratic dependence of kinetic energy on velocity. One of the purposes of the modeling was to find out if these velocity oscillations are reduced when there is elasticity in the drivetrain, since, if this is the case, this could explain the test results at least partially.

**Modeling of drivetrain with elasticity**

When the bicycle drivetrain has limited spring action, i.e. some degree of elasticity, equation (8) is no longer valid. Even when the gear ratio GR is fixed, the angular velocity of the cranks is not exactly proportional to the speed of the bicycle. In the case of the crank set of figure 1 limited elasticity is achieved by leaf springs inside the hollow crank arms. The angles of the left and right crank arms are no longer the same as the angle of the crank axle. During the downstroke energy is stored in the spring, which is then released in the dead spot in the direction of rotation. This contributes to torque on the crank axle close to the dead spot. The instantaneous applied power to the pedals is no longer equal to the power to propel the bike as a result of the temporary energy storage in the springs. Figure 7 shows the effect of the springs on the crank and crank arm angles

**Figure 7**. Deflection *θ* of the crank arm
with elasticity*,* at left during the downstroke, at right in the
dead spot

At applied torque *τ* at the pedals the crank arm deflects by
an angle *θ* :

\(\tau = F_{p}*r = k\theta\) (10)

where *k* is the spring constant of the fiber composite leaf
spring inside the hollow crank arm and *θ* is the deflection
angle of the crank arm at torque *τ* . As a result the angle
*β _{+}* of the crank arms is different from the angle of
the crank axle

\(\beta_{+} = \beta + \theta\) (11)

In the dead spot the leaf spring releases its stored energy and the
angle *θ* decreases to zero if the torque decreases to zero.

The angular velocity *ω _{+}* of the crank arm also
becomes different from the angular velocity

\(\omega_{+} = \frac{d\beta_{+}}{dt} = \omega + \frac{d\theta}{dt}\) (12)

The tangential pedal force *F _{p}* is now:

\(F_{p} = \frac{P}{\omega_{+}r}\) (13)

The energy *E _{spring}* stored in the spring is given
by:

\(E_{spring} = \frac{1}{2}k\theta^{2}\) (14)

The power *P _{spring}* is added to

\(P_{spring} = - \frac{dE_{spring}}{dt} = \ - k\theta\frac{d\theta}{dt}\) (15)

The force exerted by the spring is:

\(F_{spring} = \frac{P_{spring}}{\omega_{+}r}\) (16)

Using equations (3), (10), (15) and (16) it can be shown that:

\(F_{spring} = {- F}_{p}*\frac{P_{0}sin(2\omega t)}{\omega_{+}k}\) (17)

The force *F _{ax}* applied to the crank axle is:

\(F_{ax} = F_{p} + F_{spring} = F_{p}(1 - \frac{P_{0}\sin(2\omega t)}{\omega_{+}k})\) (18)

When only the crank arms have elasticity and the rest of the drivetrain is inelastic, the forward force applied to the bicycle, becomes:

\(F_{bike} = \frac{F_{ax}*r}{R*GR}\) (19)

Very high values of *k* represent conventional crank sets with
little elasticity, when *F _{ax}* is almost equal to

**Figure 8**. Calculated *F _{bike}*,
velocity and angular velocities vs. time for the drivetrain with
elasticity

**Calculation of speed-power ratio**

The ratio of average speed to input power was calculated at constant pace for inelastic and elastic drivetrains. Although the angular velocity profile of the crank arms changes significantly when elasticity is introduced, as shown in figure 8 , we do not see much change in the speed-power ratio at constant pace, certainly not enough to explain the experimental results. Bicycle velocity oscillations do not change much, when elasticity is introduced into the drivetrain. Therefore they cannot explain the observed test results, unless there are significant delays in the energy release of the leaf springs. This was not considered yet, because the resonance frequency of the springs is much larger than the pedaling frequency. In the case of loss-free crank sets, only a large reduction of oscillations in the bicycle velocity would reduce power input. This large reduction is not seen in the modeling to date. More accurate modeling, considering the delay in reaction to forces on the springs, may give only slightly different results.

**Losses in conventional crank sets**

The modeling above shows that speed/power ratio does not significantly change when elasticity is introduced in the drivetrain. This is attributed here to the assumption that there are no losses in a conventional crankset. It is, however, well known that during large force on the pedals during the downstroke the bicycle frame can be flexed significantly in a lateral direction (ACT Lab, 2017). In the dead spot the frame flexes back but the energy used in flexing is not returned to effective torque on the crank axle and therefore lost for propulsion. Similarly, the load on the pedal several centimeters away from the crank arm causes twisting of the crank arms (ACT Lab, 2016) during the downstrokes. Again, the energy used for twisting is not recovered in the dead spot and therefore lost.

The strain energy losses from twisting in conventional crank arms can amount to more than 1.6 % at a cadence of 100 rpm (Fairwheel Bikes, 2021) and even more at lower cadence for the same power input. We argue that in our crank set with leaf springs the strain energy on the crank arms is mostly absorbed in the leaf springs before flexing the frame and twisting the crank arms. It is then returned without significant losses in the dead spot.

**Modeling of drivetrains with loss**

To simplify modeling the flexing of the frame and twisting of the
crank arms during the downstroke are combined in the degree of
deformation *D* :

\(D = \frac{\tau}{q}\) (20)

where *τ *is again the torque applied to the crank arms and
*q* is the combined constant for flexing the frame and twisting
the crank arms. The energy *E _{def}* stored during this
deformation is, analogous to equation (14), given by:

\(E_{def} = \frac{1}{2}qD^{2}\) (21)

The total power applied to propel the bike, *P _{bike}*
, is then:

\(P_{bike} = P + P_{spring} + P_{def}\) (22)

When the strain on the frame and crank arms is reduced approaching
the dead spot, *P _{def}* is zero, because this strain
energy is not returned into effective torque on the crank axle:

\(P_{def} = - qD\frac{dD}{dt}\) if
*dD/dt* > 0 (23)

\(P_{def} = 0\) if *dD/dt*
< 0 (24)

This means that *P _{def}* is always negative or zero
and therefore reduces the power applied to propel the bike. We argue
that the presence of the springs in the crank arms significantly reduces
the flexing of the frame and the twisting of the crank arms, as compared
to a conventional crank set. In other words, the deformation of the
springs, which is reversible and contributes to propulsion in the dead
spot, replaces, at least partially, the deformation of frame and crank
arms. Irreversible losses are replaced by reversible losses.

For the conventional crankset *P _{spring}* = 0, so it
does not have the elasticity to absorb the same losses. For example, if
he deformation energy

**Conclusions**

Test results have been presented indicating that a bicycle drivetrain
with elasticity in the crank arms can improve cycling efficiency, as
reported by others as well (Bastianelli *et al*., 2019). For the
conditions used in one particular test, an increase of 2.3 % in the
speed-power ratio of our bicycle with a crank set including leaf springs
was measured. In a 45 minute time trial this would translate into more
than one minute advantage, a large improvement compared with the more
incremental benefits from aerodynamic optimization.

Our computer modeling to date cannot explain the test results unless energy losses are assumed in conventional crank sets from flexing the bicycle frame and twisting the crank arms during the downstroke. We argue that in a drivetrain with limited elasticity these irreversible losses during the downstroke are reduced and partially converted into reversible losses in the leaf springs. More testing is recommended on elastic drivetrains in general to verify and confirm our test results, as well as measurements of cranksets with elasticity in the spider (Hamamoto, 2019). Elasticity in the drivetrain can also be introduced by using stretchable drive belts instead of chains.

Other explanations than presented here are possible based on more advanced computer modeling. FEA modeling with all the material constants involved may be needed to obtain more quantitative results.

ACT Lab (2016), Crank Fatigue Test, https://www.youtube.com/watch?v=7rZ1L6brkNE

ACT Lab (2017), Frame Fatigue with Pedalling Forces Test, https://www.youtube.com/watch?v=8blo6O_KIAs

Bastianelli, B. M., Workman, A. & McGregor, S. (2019, June), Novel Crank with Elastomer Spring Improves Effective Power in Trained Cyclists and Triathletes,

*Medicine & Science in Sports & Exercise*, Chicago, 2019,

den Boer, W. (2019-1), US Patent 11,142,281, Cycle Crank Assembly, www.huroncycling.com

den Boer, W. (2019-2), White Paper IMPACT power meter August 2019: www.huroncycling.com

Fairwheel Bikes (2021), https://blog.fairwheelbikes.com/reviews-and-testing/road-bike-crank-testing/

Hamamoto, Y. (2019), US Patent 10,450,030, Rotational Apparatus and Bicycle provided with same, www.free-power.jp

Submitted by1 Sep 2023

Willem Den Boer

Huron Cycling LLC

- License: CC BY
- Review type: Open Review
- Publication type: Conference Paper
- Submission date: 1 September 2023
- Publication date: 7 September 2023
- Conference: The Evolving Scholar - BMD 2023, 5th Edition

Den Boer, W. (2023). Improvement of Cycling Efficiency for Drivetrains with Elasticity. The Evolving Scholar - BMD 2023, 5th Edition. https://doi.org/10.59490/64f21eec0b23ace23fac5785

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