Suspension Link Ratio
Link Ratio Effect On Damping:
Variable link suspension systems on motorcycles has developed into a complex, confusing and heavily covered topic. Definitions of link ratio, leverage ratio, travel ratio or motion ratio are murky with no clear discussion on how those parameters effect suspension performance. The attempt here is create some easily measured simple intuitive parameters. Figure out the effect of those parameters on suspension performance and how to go about tuning linked suspensions to control suspension performance.
How To Figure Out The Link Ratio Of Your Suspension
Travel ratio (TR) and Link ratio (LR) are straight forward to calculate given four or more measurements of shock position over the rear wheel stroke. The first step is plot the shock versus wheel position measurements in excel (or open office) and use the curve fitting features of the spreadsheet to fit the data. Steps to curve fit the data are shown below.
It is important to understand spreadsheets will fit the data you feed it with impunity. Any data that is a little bit off, will drive the curve fit off and create a convulsion instead of a fit. More data points will give a better curve, any data that falls off of the curve is worth double checking and throwing out if known to be bad.
Get through the curve fit process and excel gives you a polynomial equation describing the shock shaft position as a function of wheel travel. Those polynomial coefficients (b,c,d) are used to compute travel ratio (TR) and link ratio (LR) using the equations below. With the equations you can calculate the shock position and link ratios over the stroke in 1 mm increments or any other wheel position you want.
Spring Force At Wheel
To get spring force at the wheel the first step is figure out how far the spring is compressed when the wheel/suspension is moved to some stroke position specified by y.wheel. Spring position is straight forward to calculate using the travel ratio (TR) equation.
Given how far the shock spring has been compressed (y.spring) the spring force is calculated by the spring stiffness constant and the initial preload on the spring (y.preload).
Force amplification through the link system to the rear wheel is determined by a simple teeter-totter torque balance. The link ratio (LR) specifies how far the spring moves for an incremental change in wheel position. The link ratio also specifies the teeter-totter force transfer through the suspension link to the rear wheel.
Combining all the above into a single equation gives a relationship specifying rear wheel spring force at the suspension stroke position y.wheel.
For an incremental change in stroke position the spring rate at the rear wheel is:
Damping Force At The Wheel
To figure out damping force generated by the shock the first step is to determine the shock shaft velocity given the rear wheel suspension velocity. The link ratio (LR) gives that relationship.
Damping force produced at the shock shaft is defined by the shock damping coefficient and shaft velocity.
Damping force transferred to the rear wheel is determined by the same link ratio (LR) teeter-totter torque balance used for spring force. Combining all the above gives a single equation for damping force at the wheel as a function of wheel bump velocity. For suspension response calculations that equation is rewritten in terms of the rear wheel damping coefficient c.wheel defined in terms of the link ratio (LR) and the shock damping coefficient c.shock.
Link Ratio Effect On Suspension Response
On a fork, spring force and damping force act directly on the wheel. That makes calculation of the suspension response coefficients tau and zeta straight forward.
To get the actual forces acting on the wheel in a linked suspension system, the travel ratio and link ratio functions are needed to translate forces at the shock to the actual forces acting on the wheel. Adding those functions to the tau and zeta equations creates two terms: LR.fac acting on the zeta equation and LK.tau acting on the tau equation.
Suspension link ratio effects on suspension response coefficients tau and zeta are all lumped into two parameters LR.fac and LK.tau. Those parameters are function of travel ratio (TR) and link ratio (LR). Both parameters are direct multipliers on the suspension response coefficients tau and zeta which means link ratio effects suspension response independent of shock damping or spring stiffness.
Zeta Effect On Suspension Performance
An increase in spring stiffness (k.spring) reduces the value of zeta and drives the suspension toward an under damped condition. Likewise reduced shock damping creates a lower damping coefficient (c) and reduces the value of zeta. Weight, spring rate or damping all effect the value of zeta and that has a direct impact on suspension response as shown in the curves below.
Low values of zeta (like 0.3) get the wheel over the top of a bump faster (plush) but the suspension then goes into a baby-buggy oscillation after the bump. Higher values of zeta (like 0.7) damp the oscillation faster but require longer to clear the top of a bump - that means the bump is pounding directly into the bars because the wheel can not get out of the way. Performance suspensions are tuned around zeta values of 0.5 to 0.7. Passenger cars run around 0.3.
Link Ratio Effects On Suspension Damping Performance
Linked suspensions provide a way to change the "effective" spring stiffness at the wheel by changing the link ratio. The link ratio also effects the shock shaft speed which in turn effects damping force generated by the shock. The LR.fac parameter combines all of those effects into a single parameter and defines the effect of link ratio modifications on suspension response. As a direct multiplier on zeta, a 10% increase in LR.fac creates a 10% increase in zeta.
The plots below show examples of link ratio modifications. Increasing link ratio decreases stability at the bikes normal ride height and then increases stability as the suspension is driven deeper in the stroke. That behavior is easy to figure out in the LR.fac curve (far right), but not obvious in the travel ratio curve (center).
The link ratio gives a more intuitive result on the effect of changes. Link ratio is defined as the change in shock shaft position for an incremental change in wheel position. The link ratio curve shows less shock motion around the normal ride height so it is not surprising the suspension is less stable there. Deeper in the stroke the link ratio increases, increasing the shock shaft velocity and damping force, making the suspension more stable deeper in the stroke.
The link ratio curve more closely follows the LR.fac curve and gives a more intuitive result. But it is important to remember LR.fac is a function of both travel ratio and link ratio and it is the value of LR.fac that determines suspension response through it's effect on zeta. Link ratio or travel ratio alone are insufficient - you need the LR.fac value.
Link Ratio Effects On Damping Performance
LR.fac values on a bike-by-bike basis are all over the place, but there are some generalities between street and longer travel dirt bike suspensions:
Every bike is a little different and the only way to get exact values is measure them. The differences between street bikes and dirt bikes divide link ratios into three different groups.
Track bikes and sport bikes are designed for smooth high speed tracks. Suspensions on those bikes don't have much travel or progression in link ratio over the stroke. The LR.fac progression is about 30% for the bikes shown. Starting from the race sag position the increase in LR.fac is about 20% at bottoming. That means a bike tuned for a suspension response coefficient (zeta) of 0.7 at normal race sag would hit a value of 0.84 (20%) as the suspension taps bottom. Keeping the link ratio progression small gives consistent suspension response and feel over the stroke making tuning easier.
Street bikes are a different deal. At race sag link ratios (LR.fac) are about the same as track bikes ~ 0.5. That means the spring rate and race sag will be about the same as track bikes. As the suspension is driven deeper in the stroke the link ratio (LR.fac) increases by a factor of two at end of stroke. That aggressive progression lets the suspension run around in an under damped condition at the normal ride height to give a smooth/compliant ride. Hit a bump and LR.fac increase as the suspension is driven into the stroke causing the suspension damping response coefficient (zeta) to increase by a factor of two. That converts what was an under damped suspension into something closer to critically damped. Practical, but tuning can be awkward because the suspension response zeta coefficient changes on a bump-by-bump bases depending on how deep the suspension is driven into the stroke.
The link ratio LR.fac curve starts to kick up at around 50% of travel. From the normal ride height, 30% travel, the suspension can stroke through another 20% and stay in the linear range. Suspension strokes deeper than 50% are in the aggressive LR.fac increase range for bottoming control.
Dirt bikes and tarmac track bikes have a surprising similarity in link ratios. Over the first six inches of stroke, which is all a track bike has, the progression in link ratio is about the same for both bikes at 30%. Through that range shock tuning controls bumps and chassis damping when the suspension operating around the normal ride height in the first six inches of travel.
Drive deeper in the stroke and the bottoming system kicks in on a dirt bike at around 70% travel. Progression in link ratios makes the final travel stiffer in both spring force and damping. The damping increase at end of stroke is important to remember. If you start out with a suspension close to critical damping at the normal ride height the suspension will be driven into an over damped condition by the increase in LR.fac at end of stroke.
Linked Suspension Tuning Approach:
At race sag link ratios and LR.fac values are pretty linear and do not change much through the first 50% of travel. Through that range shocks are tuned to handle normal bumps and chassis damping. Drive deeper in the stroke and the bottoming system kicks in with an aggressive increase in link ratio. The end of stroke link ratio increases both spring force and damping. Evaluating link ratios in terms of LR.fac shows the link progression increase damping more than spring force driving the suspension towards an over damped condition at end of stroke.
The progression in LR.fac is important to keep in mind when tuning. An example where link ratios can cause a problem is re-spring a bike for a lighter rider. Reducing the spring rate and rider weight increases the value of zeta which drives the suspension towards an over damped condition. That may work OK at the normal ride height, but when the suspension is driven to end of stroke on a dirt bike the link ratio doubles and that doubles the value of LR.fac. Since LR.fac is a direct multiplier on zeta the increase in LR.fac doubles the value of zeta making the suspension over damped. That causes problems in whoop sections where repeated bottoming will cause the suspension to pack. The solution is re-tune the shock damping coefficient (c.shock) for the change in rider weight and spring rate to restore the stock zeta value at the suspension normal ride height.