Computation spring rate

Preload

Natural frequency undamped

Damping

Pogo by bumps in the road

Pogo by pedal force

Losses. A simple pogo-model

Estimating damping losses

Theoretical calculations

Measurements

Conclusions

Links

E-mail: Gert van de Kraats

Elsewhere, it has been calculated how to minimize pogo or squat from the rear suspension of a recumbent bike.

An obvious question, however, is how much squat is created if not minimized and how much energy this actually takes. Especially uphill at low gear and cadence, squat can become a nuisance and give the feeling that a lot of energy is wasted.

We want to know what the influence of the selected gear and pedaling frequency is on squat behavior and energy loss, but this question does not seem easy to be answered.

To keep the calculation somewhat simple, we assume that the spring is mounted vertically and we neglect the angle of the rear fork to the horizon.

It is also assumed that energy loss in frame and suspension is small compared to the total pedaling power.

There are 3 possible causes of losses in suspension :

- spring elements are provided with damping, which must prevent undesirably large spring deflections. This damping always costs energy, if the spring length changes.
- during a pedaling movement, with increasing pedaling force, part of the force is used to push or pull out the spring. The question is whether the energy that this costs is released again when the pedaling force decreases .
- possibly, storing energy in the spring and releasing that energy again later, leads to a less efficient use of muscle power.

At Bicycle Suspension of Walter Zorn (this Java-program at this moment only works under Windows 10 with Internet Explorer) the relationship between spring rate and position of the spring on the rear fork is graphically displayed.

The spring rate is often given in lbs/inch, converted to the metric measure Newton/meter:

1 inch = 2.54 cm.,

1 lbs = 4.45 N,

total conversion factor: 4.45 * 100 / 2.54.

In the calculation it is important which percentage of the total weight presses on the rear wheel.
The position of the spring on the swing arm determines the factor by which the force due to the total weight must be multiplied to calculate the force on the spring.
The spring rate of the spring must now be selected so that the spring is compressed 20 to 30% of the maximum deflection when the rider is on the bicycle.
This setting ensures that 80 to 70% of the spring travel can be used for sudden compression of the spring (a bump), and 20 to 30% for ejection (a hole).

This is a logical choice.
If a wheel is not pressed into a deep hole at the bottom, this only means that the wheel will spin when you pedal.
If, on the other hand, the maximum spring deflection is too small to follow a large bump, the bicycle and rider receive a strong blow and are launched.

Formules:

Gravity at the rear axle:

F_{a} = m * 9.8 * L_{cg} / L_{wb}

= m * 9.8 * perc_{a} / 100

where L_{cg} is the distance from the center of gravity to the front axle and L_{wb} is the length of the wheelbase.

The force on the spring:

F_{v} = F_{a} * L_{a} / L_{v}

where L_{a} is the length of the rear fork and L_{v} the distance from the spring to the pivot point.

The spring rate:

k = F_{v} * 100 / (x_{v} * perc_{v})

where x_{v} is the maximum suspension travel and perc_{v} is the percentage of the spring travel (sag) that the spring element is compressed under load.

It is remarkable that the maximum spring travel and the damping of a spring element can often not be found in the specifications

Spring elements are provided with the option to "preload" the spring.

This ensures that the spring element is pressed in less ("sag") when the rider sits on the bicycle.

A common setting is 30% sag, where the spring element is depressed 30% when the rider sits on the bike.

The "preload" ensures that the down bounding spring travel is increased at the expense of the up bounding spring travel.
This can be used to correct a too weak spring or a heavily loaded bicycle.

However, this has no influence on the other properties such as damping and resonance.

Guus van de Beek shows in Ligfiets& 2003-1 that the spring deflection of the rear spring does not only depend on the force, but also on the speed at which the force increases/decreases, i.e. on the frequency of the force.

The concept of **natural frequency** is important here.
If an undamped spring with a weight attached to it is pressed once, the weight will continue to sway for a long time (free vibration) at a frequency that depends on the mass and the stiffness of the spring.
This frequency is called the natural frequency.

The suspension system of a recumbent bicycle only functions properly if the frequency of the unevenness and associated forces are well above the natural frequency.
If the frequency is equal to the natural frequency, the spring system is malfunctioning. Resonance occurs, leading to large spring travel and a violently swaying bike.

This phenomenon is shown graphically for a general mass-spring system at Forced oscillations (resonance) from Walter Fendt.

The **natural angular frequency** ω_{0} (in radians/second) of a mass spring system is determined via the formula

ω_{0} = sqrt(k / m),

where k is the spring rate and m the mass.

One problem is that mass and spring are not directly coupled to each other, but via a double lever construction.
The frame and rider "hinge" about the front axle; rear fork and rear wheel "hinge" around the suspension point of the rear fork on the frame.

To make a calculation, we replace the "real" spring with an "imaginary" spring, which is positioned directly on the center of gravity. This imaginary spring should give the same spring properties to the recumbent bicycle as the real spring.

The "leverage ratio" LR is important here; this is the ratio between the length change of the spring and the height change of the center of gravity.

The ratio between the real spring constant and the imaginary spring constant is determined by the formula:

k / k_{i} = LR^{2} .

We determine LR in 2 steps. We first move the spring to the rear axle and then from the rear axle to the center of gravity.

If we move the spring to the rear fork, the length change of the spring increases directly proportional to the length of the rear fork. The following applies to this leverage ratio:

LR_{a} = L_{a} / L_{v} .

In the calculation we assume that the total mass of the recumbent bicycle and cyclist is located at the center of gravity. The center of gravity on a high recumbent is located midway between the front and rear wheel.
With a suspension rear fork, this center of gravity pivots around the front wheel. If the spring is moved to the center of gravity, this means that the deflection that the spring has to make decreases proportionally with the distance from the front axle. The following applies to this leverage ratio:

LR_{cg} = L_{cg} / L_{lwb} .

Since the ratio between the position of the center of gravity and the total wheelbase is related to the percentage of the weight pressing on the rear wheel, it can also be written as:

LR_{cg} = perc_{a} / 100.

The following applies to the total leverage ratio:
LR = LR_{a} * LR_{cg} = (L_{a} / L_{v}) * (perc_{a} / 100) .

The **natural angular frequency** of imaginary spring with recumbent and rider is now:

ω_{0} = (sqrt(k_{i} / m)).

After filling in k_{i} = k / LR^{2} this leads to:

ω_{0} = (sqrt(k / m) * (L_{v} / L_{a}) * (100 / perc_{a}).

The **natural frequency** is determined by dividing this angular frequency by 2π.

The number of pedal revolutions/minute is determined by multiplying by 30 (2 times force is applied for every pedal revolution).

A spring is provided with damping to prevent resonance or prolonged deteriorating of the spring in the event of an unevenness in the road.
Although the damping of a spring is very important for the spring properties, the damping factor is (almost) never mentioned for unclear reasons!?

Different names and symbols are also used for the definition of damping.
There is talk of "damping constant" b or λ , of "damping factor" Γ (=b/m), or of "damping number" γ (=w/2m).Here the "damping constant" b is used and the formula for damping is:

damping force = b * velocity (of spring movement).

A **damped** spring with a weight attached to it will, after pressing once, decelerate, but the spring deflection will decrease quickly as a result of the damping.
The formula for the angular frequency of the damped free vibration is:

ω_{1} = sqrt(ω_{0}^{2} - b^{2}/4m^{2})),

see free vibration (Dutch from NIKHEF).

The angular frequency of a damped free vibration is lower than that of an undamped one. The angular frequency decreases as the damping increases.

At b = sqrt (4 * k * m) the angular frequency is 0. This is called **critical damping**.
With greater damping, the spring no longer bounces, but slowly returns to the equilibrium position.

In practice this means that after a bump the wheel can no longer be pushed back by the spring, so that the wheel will come off the road.

For a properly functioning suspension system, the damping must be smaller than the critical damping. Such a spring system is called 'weakly damped'. The damping must be:

b < sqrt (4 * k * m) .

The relationship between the damping constant of the real spring and the imaginary spring at the center of gravity is determined by the same formula as for the spring constant:

b / b_{i} = LR^{2} .

OTo view the effects of a bumpy road on the suspension behaviour, the massless side of the spring is moved up and down via a **forced** vibration (through bumps and potholes in the road).
We represent this forced oscillation with a cosine function:

y(t) = y_{0} * cos(ω_{d}t)

where ω_{d} represents the angular frequency of the bump motion and y_{0} represents the maximum **size** of the bump.

For the vertical spring travel this gives the following formula:

y(t) = (y_{0} * k / m) * cos(ω_{d}t +δ) / sqrt(b^{2}ω_{d}^{2}/m^{2} + (ω_{0}^{2}-ω_{d}^{2})^{2} )

where b represents the damping constant of the spring,
and δ the phase shift.

The spring deflection is maximal at the resonance angular frequency:

ω_{R} = sqrt(ω_{0}^{2} - b^{2}/2m^{2})).

This formula is derived from formulas at Forced oscillations (resonance) by Walter Fendt and at HOBBING, SWINGING AND VIBRATION (Dutch) from UTwente.

Here too there is a **forced** vibration. Not because of differences in height, but because of a varying force.

In standard calculations on forced vibrations it is assumed that the force flows according to a cosine function:

F(t) = F_{0} * cos(ω_{d}t)

See e.g. graphs, formulas and derivations (from NIKHEF (Dutch)) about free and forced vibrations

or formulas at https://www.jirka.org/diffyqs/html/forcedo_section.html

The following formula can then be used for the vertical spring travel:

y(t) = ((F_{0}/m) / sqrt(b^{2}ω_{d}^{2}/m^{2} + (ω_{0}^{2}-ω_{d}^{2})^{2} )) * cos(ω_{d}t +δ)

where b represents the damping constant of the spring,
and δ the phase shift.

The maximum spring deflection is therefore:

y_{max} = (F_{0}/m) / sqrt(b^{2}ω_{d}^{2}/m^{2} + (ω_{0}^{2}-ω_{d}^{2})^{2} )

As with "bumps in the road", the spring travel is maximal at the resonance angular frequency:

ω_{R} = sqrt(ω_{0}^{2} - b^{2}/2m^{2})),

where the damping b must satisfy the condition:

b <= sqrt (2 * k * m) .

With greater damping, the resonant frequency becomes lower, but the maximum spring deflection at that frequency also decreases.

If the damping is chosen greater than sqrt (2 * k * m), (but smaller than sqrt (4 * k * m), so still 'weakly damped'), then the spring deflection is maximum at cadence 0 and the spring deflection increases at greater frequency only.

The next formula applies to the average power supplied:

P = (F_{0}^{2}/2b) * (b^{2}ω_{d}^{2}/m^{2}) / ((b^{2}ω_{d}^{2}/m^{2}) + (ω_{0}^{2}-ω_{d}^{2})^{2})

It is easy to see that according to this formula at ω_{d} = ω_{0} the greatest power is delivered to the damper:

P = F_{0}^{2}/2b with a spring deflection:

y_{max} = F_{0}/(bω_{0})

This corresponds to the general relationship between power and spring deflection at a given angular frequency:

P = b ω_{d}^{2}y_{max}^{2} / 2

The (pogo) force of the pedaling cyclist does not change direction during a complete pedaling movement, but is, depending on the setting of the bicycle, either continuously positive or continuously negative. We therefore approximate the (pogo) force (very roughly) by this cosine function:

F(t) = F_{0} * (1 + cos(ω_{d}t)) = F_{0} + F_{0} * cos(ω_{d}t)

where ω_{d} is the doubled value of the angular frequency of the pedaling movement and F_{0} is set equal to the vertical pogo percentage of the average pedaling force. The doubling of ω_{d} is caused by the fact that 2 times force is applied per full circular pedal rotation.

The constant first term F_{0} and the constant gravity have no influence on the resonance behavior of the mass spring system.

The average power output does not change due to the additional term F_{0}. However, the maximum spring deflection must be multiplied by a factor of 2.

Note, the above formula assumes that the force acts on the mass.
The vertical pogo percentage calculated elsewhere is based on a force at the rear axle. As a result, the vertical pogo force must be multiplied by a factor L_{lwb} / L_{cg}, to calculate F_{0}.

The energy, which is added to the mass spring system as potential energy or kinetic energy with increasing (pedaling) force, is completely released again, apart from damping losses, with decreasing (pedaling) force.

By using a simple pogo model, it is made clear what happens to this energy in a bicycle with rear suspension.

For an introductory explanation, see Pogo (squat) at a recumbent, where the basis for this model is being laid.

The pedaling force creates a horizontal forward force F_{d} at the contact point of rear wheel with the road, which propels the bicycle. When all the forces of rear fork and front fork caused by pedal force on the frame are added together, this results in a horizontal forward force on the frame.

In the graphic simulation the horizontal "blue" arrow above the front axle shows this force on the frame.

If this horizontal force is not directed through the center of gravity ("cg"), the frame will rotate. Above the front axis the frame cannot move vertically and will therefore rotate about a point directly above the front axis, at the heigth of the center of gravity.

This is shown below in the simple pogo model of a bicycle with rear suspension and negative pogo.

The magnitude of the force F_{d} depends on the bicycle configuration.

With negative pogo, the force is positioned lower than the center of gravity (as in the drawn model) and the rear suspension will compress as soon as force is applied.

In positive pogo, the force is positioned higher than the center of gravity and the rear suspension will bounce as soon as force is applied.

Ideally, the power runs through the center of gravity and there is no pogo.

The figure below shows the horizontal force F_{dx}, which is caused by the increasing horizontal (pedal) force F_{d} and runs through the center of gravity. F_{dx} increases the frame speed.

In the absence of pedaling force, gravity and spring force are in equilibrium and there is no vertical force F_{cgy}.

An **increasing** kicking force causes a **down** force F_{cgy}. This causes the **inflection** of the spring.

Not all the energy of the pedaling force F_{d} is used to increase the speed, but part of the energy is used for this compression of the rear suspension and is stored as potential energy in the spring.

The compression ensures that the front axle travels a greater distance than the center of gravity and that more power is also delivered than without compression.

When the pedaling force has reached its maximum and the pedaling force **decreases**, the spring force becomes greater than the vertical moment force and F_{cgy} will be **upwards** directed, as shown in the figure below. This causes the **bounce** of the spring.

The front axle travels a shorter distance than the center of gravity and less pedaling power is delivered than without suspension.

The forward force F_{dx} is also in this case the same as F_{d}, but is exerted over a greater distance. This provides more forward power at the center of gravity than the pedaling power delivered. In this phase, the potential energy released from the spring system is used to accelerate the bicycle.

The compression itself therefore does not cost any power, only the damping causes loss.

For a calculation model see Estimate losses (Dutch), where the influence of different recumbent parameters on the damping losses is calculated.

The parameters used are based on the Nazca Pioneer.

We assume that the spring behavior of the spring and damper is optimal if the spring deflection at the resonance frequency is not greater than 2 times the deflection at a very low frequency (3 dB damping).

Voor b moet dan gelden:

b = sqrt ((1 / 4) * k * m) .

The proposed damping constant is therefore based on a '3 dB weakly damped' system and is calculated using the above formula. In the empty entry box you can enter a damping constant (b), which then replaces the proposed damping constant in the calculation.

The deflection and the pogo loss are calculated based on the specified Cadence and Pedal Power.

The graph shows the deflection and the power loss as a function of the cadence.

Very few publications are known in which an attempt is made to calculate suspension losses due to pogo. The first publications only appear with the advent of mountain bikes with front and rear wheel suspension.

Eric L. Wang and M. L. Hull publish in 1996 and in 1997 Minimization of Pedaling Induced Energy Losses Rear Suspension Systems
.
The losses are calculated here by means of a simulation on an imaginary bend bicycle, in which the height of the pivot point can be varied. In the simulation, the bicycle is driven by pedaling force, which is measured on a comparable, but **unsprung** bicycle.

The simulation bike rides on a conveyor belt with a 6% incline, which will cause pogo effects for squatting.

On a commercial bicycle, the losses in the suspension are 6.9 Watt at a pedaling power of 531 Watt (1.3%). At an optimal height of the swingarm pivot point, the calculated losses are only 1.2 Watt (0.2%).

It is not clear how these losses were calculated. It seems that only the losses in the spring element have been taken into account, which are caused by the damper and by friction.

It is remarkable that the calculated optimum height of the hinge point is well above the chain line (factor 2 compared to the used front sprocket). This contradicts the model on this website.

The measurement bike used for verification of the simulation results is described in An Off Road Bicycle With Adjustable Suspension Kinematics. The results of the measurements on this measuring bike show a much lower optimal pivot point, which is still slightly above the chainline./p>

Ari Karchin and M. L. Hull publish 2002 measurements in of losses at different swingarm pivot heights.

In these measurements, the optimum pivot point for the **seated** cyclist is slightly above the chainline, the optimum pivot point for the **standing** cyclist is below the chainline, as in the simulation model of this website (see model mountain bike Giant Reign).< br>
Here too it is unclear how the losses are determined.

The measured losses are much larger than the losses calculated on this page.

A good suspension system ensures a natural frequency that is well above the optimal cadence of 90 revolutions/minute.

A slacker spring (with a smaller spring rate) results in more sway and an increase in damping loss; however, the maximum damping loss (at the lower resonant frequency) does not increase.

The power loss due to swell is proportional to the square of the force and inversely proportional to the damping constant. A choice for greater damping therefore results in a decrease in power loss (but for poorer spring properties).

With low damping, the spring deflection is greatest at the resonance frequency.
With greater damping, the maximum spring deflection decreases and also occurs at a lower cadence.

The greatest power loss due to pedaling force does not occur at the greatest spring deflection, but occurs at the natural frequency.

Given the importance of the damping constant and also of the effective spring travel for the behavior of a spring element, it is remarkable that manufacturers (almost) never mention these characteristics in the specifications.

In practice, the pogo losses will be greater than calculated here, because the pedaling force does not follow a cosine function neatly.

Nevertheless, we can conclude that with a well-sized suspension system and a good chainline, the pogo losses are negligible.

The effects of suspension on the energetics and mechanics of riding bicycles on smooth uphill surfaces, by Asher H. Straw, 2017.

A Three-Dimensional Multibody Model of a Full Suspension Mountain Bike, by Burkhard Corves
, J. Breuer, Frederic SchÃ¶ler, P. Ingenlath, 2015.

Bicycle Shock Absorption Systems
and Energy Expended by the Cyclist, by Henri Nielens and Thierry Lejeune, 2004.

A Multibody Model for the Simulation of
Bicycle Suspension Systems, by MATTHIAS WAECHTER, FALK RIESS and NORBERT ZACHARIAS, 2002.

Experimental Optimization of Pivot Point Height for Swing-Arm Type Rear Suspensions in Off-Road Bicycles
, by Ari Karchin and M. L. Hull, 2002.

Minimization of Pedaling Induced Energy Losses in Off-road Bicycle Rear Suspension Systems
, by Eric L. Wang and M. L. Hull, 1997.

An Off-Road Bicycle With Adjustable Suspension Kinematics
, by S. A. Needle and M. L. Hull, 1997.

A Model for Determining Rider Induced Energy Losses in Bicycle Suspension Systems
, by Eric L. Wang and M. L. Hull, 1996.