## Definitions

### Coordinate Systems

##### Bike Coordinate System (Static)

The bike (or vehicle) coordinate system treats the frame as a static reference point from which forces and displacement are measured. The **Frame Reference** selection in the **Bike Plot** tab visualizes this arrangement. The frame (front triangle) is fixed in space and the unsprung components move with respect to the frame. Static metrics are taken in this frame of reference.

##### Ground Coordinate System

The ground coordinate system fixes the wheels to the ground and allows the frame to rotate and move as the suspension moves. The rider COG is treated as fixed with respect to the frame and will move and rotate with the frame. Forces are taken with respect to the ground place and so a vertical force will not always be vertical with respect to the front triangle. The **Ground Reference** selection on the **Bike Plot** tab will visualize this.

This frame of reference much more accurately represent real world bike behaviour as the suspension forces into the bike are dominated by the tire contact patch to ground relationship.

### Kinematics

##### Leverage

Proportion of vertical wheel force to shock force in the bike coordinate system.

Vertical leverage does not account for the bike sagging into itself throughout the travel and always treats the force as vertical with respect to the bike coordinate system.

Can also be defined as the instantaneous vertical wheel movement proportional to the instantaneous shock movement.

Lev_{vertical} = F_{wheel vertical} / F_{shock}

##### Leverage Horizontal

Proportion of vertical wheel force to shock force with respect to the ground coordinate system. As the bike sags into itself the force direction moves around to the wheel to always be vertical with respect to the ground.

Lev_{horz} = F_{wheel vertical*} / F_{shock}

##### Leverage Square Edge

Proportion of force from a square impact at a given height to shock force. As the wheel hits a square edge the force application will not be vertical to the ground but radial to the wheel at the contact point. This is a good indication of the wheels ability move around an object.

Lev_{horz} = F_{bump contact point} / F_{shock}

##### Antisquat and Rise

The anti numbers are the proportion between the amount of compression or extension the suspension would normally move for a given vertical force and the amount the suspensions moves for the same force due to weight transfer. Due to the way the acceleration forces are applied there is usually additional forces that increase or decrease the suspension movement.

Standard and static metrics are provided. The standard measurement takes the anti values with respect to the ground coordinate system; the static measurements use the frame coordinate system and treat the frame as fixed member.

Anti % = 100% × F_{shock accel forces} / F_{shock weight transfer}

##### Progression

Progression is a relative measure of the leverage increase throughout the travel. Progression can be measured in several different ways but the most common approach uses the proportion of leverage difference from start to end to the starting leverage (or sometimes the maximum leverage)

Progression % = 100% × (Lev_{start} - Lev_{end}) / Lev_{start}

##### Progression Alt.

An alternative calculation uses the proportion with respect to the ending leverage. This can often better represent the differences in progression for very progressive bikes (over 30%).

Progression % = 100% × (Lev_{start} - Lev_{end}) / Lev_{end}

##### Wheel Force

Wheel force is the vertical force applied to the tire contact patch by the shock. This is measured in the ground coordinate system and is the shock force at a given shock travel multiplied by the leverage ratio at that travel.

When compared to leverage the wheel force curves gives a much better indication of rider support as it factors in elements such as volume spacer selection and air can properties.

F_{wheel} = Lev_{vertical*} × F_{shock}

##### Wheel Rate

Wheel rate is the instantaneous change in spring rate at a given travel point. This is a measure of how much rate stiffening occurs through the travel. Wheel rate offers similar insight as progression while also accounting for spring properties. Wheel rate is the derivative (slope; rate of change) of the wheel force curve.

Wheel Rate = dF_{wheel} / dx

##### Energy Absorbed

Energy absorbed is the spring energy absorbed by the shock for a given displacement. This is a helpful way to account for large impact absorption (bottom out protection) by accounting for the shock deceleration forces through the travel.

Coil springs tend to have higher midstroke forces and lower endstroke forces than air shocks allowing them to slow a bottom out even more early in the stroke and less later in the stroke. Energy absorption can provide a better metric for comparing bottom out protection between air and coil than progression.

Energy absorbed is given by the integral of the shock force curve or the wheel force curve.

Energy Absorbed = ∫ F_{shock} dx

##### Chain Growth

Chain growth is composed of three major components.

**Tangent Length:**The change in length of the tangent line from the two chain take-off points. Shown in blue in the figure below.**Chain Wrap Length:**The change in length of the chain wrapping around the sprocket due the difference in sprocket diameters. This is because the tangent point rotation about the sprocket axis is the same on each sprocket but the arc length change is given by angular change multiplied by the sprocket radius and so different radii will result in a different wrap length change on each sprocket. This is shown by the pink arcs on the figure below.**Wheel Wind Up:**The tire contact patch is considered instantaneously fixed to the ground as a pivot. As the axle path changes this causes the wheel to rotate and wind or unwind the chain around the cassette. If you were to test pedal kickback with the a bike in a stand and the rear wheel lifted from the ground you would see this effect as a slight rotation of the wheel. This effect is often quite minor but can become significant on bikes with very rearward axle paths.

**Note:** The angular change of the tangent points about the sprocket axis must be the same because the lines drawn from the sprocket centers to the tangent points are perpendicular to the same line by definition. As the angle of the tangent-point to tangent-point line changes these radial lines must rotate by the same magnitude.

For two equally sized sprockets there would not be a chain wrap length difference for any wheel position.

Two chaingrowth parameters are offered. **Chain Growth** is the total change in length. **Chain Growth Tangent** is the tangential length change only for reference. The total chaingrowth represents the true impact on pedal kickback.

##### Pedal Kickback

The primary pedal kickback parameter is directly calculatable from the total chain growth and the chainring radius. Because the wheel is not permitted to rotate any chain shortening must be taken up by pulling on the chainring.

Pedal Kickback = chaingrowth_{total} / r_{chainring}

**Note:** The results of this pedal kickback calculation have been experimentally verified. For more information or questions on pedal kickback and chain growth please email support@syn.bike.

A secondary pedal kickback parameter is provided to align with the commonly used Linkage X3 software. This calculation use the same angular change from the tangent length and the wheel wind but a different calculation for the result of the wrap change. *These results do not correlate well with the experimental data. *

θ_{wheel wind} = d_{axle path} / r_{tire} × r_{cassette} / r_{chainring}

θ_{wheel wrap} = (θ_{chainline} - θ_{chainline original} - θ_{wheelbase change}) * (r_{cassette} / r_{chainring} - 1)

Pedal Kickback = chaingrowth_{tangent} / r_{chainring} - θ_{wheel wrap} - θ_{wheel wind}

Where θ_{chainline} is the angle of the chainline with respect to the horizontal axis of the frame coordinate system and θ_{wheelbase} is equal to the angle between the horizontal axis of the frame coordinate system and a line tangent to the bottom of both wheels.