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Scaling OEM Damping Rates using Shim ReStackor
From spring-mass-damper theory there are two fundamental parameters that control the characteristics and feel of a suspension system. These are the suspension oscillation time constant (tau) and the suspension system damping coefficient (zeta). If two bikes with different rider weights and spring rates are setup so that these two fundamental parameters match then the bikes will have the same suspension response, feel and behavior.
The suspension oscillation time constant (tau) is defined as:
Defining tau as the square root of the mass (m) divided by spring rate (k) make tau essentially equivalent to race sag. The coefficient g.c is a unit conversion that relates the units of force and mass and is a constant. Since riders typically setup their bike to have the same race sag the time constant tau will inherently be the same.
The second parameter (zeta) is the suspension damping coefficient:
Here the added parameter is the shock absorber damping coefficient (c) which is simply the force produced by the shock divided by the shaft velocity.
The critical fact from spring-mass-damper theory is the two suspension parameters tau and zeta completely describe the oscillation characteristics of a suspension . For two riders of different weight, on different bikes, with different suspension systems as long as tau and zeta are the same the suspension travel, response and oscillation behavior will be the same.
With that in mind we can manipulate the equation for tau to determine the relationship of rider weight, spring rate and the suspension time constant. This relationship will be stuffed into the zeta equation below:
Using the relationship for zeta the value of the shock absorber damping coefficient (c) required to set zeta to be the same for both bikes is:
Damping Requirements for a Heavier Rider
From spring-mass-damper suspension theory a heaver rider requires the suspension damping rates to be increased by the ratio of the spring rate change. Use of the baseline Shim ReStackor code to scale the suspension damping forces requires a way to estimate the relationship between the shim stack flow area computed by Shim ReStackor and the suspension damping coefficient (c). That relationship can be estimated using the Bernoulli equation.
In the above equation c.d is the discharge coefficient for the shock absorber valve and will be the same for both bikes as long as the valve geometry is not modified. The shock absorber fluid density (rho) and viscosity is assumed to be the same for both bikes as well.
On any given bump we want the suspension response to be the same and produce the same suspension travel. That means the fluid flow rate through the shock absorber valve (w) will be the same. With this in mind we can lump all of the parameters that will be the same for both bikes: mass flow (w); discharge coefficient (cd) and fluid density (rho); into a single parameter (w') and set (w') to be the same for both bikes:
Applying the Bernoulli equation shows we need to setup the shim stack stiffness for the heavier rider so that the shim stack flow area is proportional to the square root of the pressure difference we want in the shock. The pressure difference across the shock absorber valve is the damping force produced by the shock. This is the relationship we need for scaling the damping force.
Returning to the zeta equation from spring-mass-damper theory we can now set the damping force needed for the heavier rider. For the special case where the suspension velocities are the same for both bikes we can use the zeta relationship to define the difference in damping force, valve pressure drop and shim stack flow area needed for the heavier rider:
To first order, the stiffness of the stack is proportional to the force applied to the stack divided by the flow area produced.
From the above example at a fixed suspension velocity the ratio of damping force and stack flow area is:
Substituting these relationships into the equation for stack stiffness gives the ratio of the stack stiffness as:
At the same suspension velocity the flow area of the modified shim stack will be proportional to the square root of the spring rate change. At the same valve pressure drop, and different suspension velocities, the flow area of the shim stack will be directly proportional to the ratio of the spring rate change.
The fundamentals of spring-mass-damper theory provide the relationships needed to determine the shim stack stiffness needed to damp the suspension for a rider of different weight. The capability of Shim ReStackor to accurately compute the stack face flow area of a shim stack gives you the capability to scale the damping rates of a stock suspension, and the rider weight it was designed for, to your weight and use the fundamental of spring-mass-damper theory to maintain the suspension response, feel and performance the manufacturer intended your bike to have.
For an example of application of this process check out the crf450r example in the sample applications section of the ReStackor web site.
Weight Scaling Damping Rates of the Stock Suspension for a Rider of Different Weight
From the fundamentals of spring-mass-damper theory there are two parameters the control the suspension response. Tau defines the oscillation time constant of the suspension and will be the same for two riders of different weight if the bikes are setup to have the same race sag. The zeta coefficient describes the damping characteristics of the suspension. To maintain the same damping characteristics when the spring rate and rider weight are changed the shim stack stiffness must be changed so that the ratio of stack flow area at the same valve pressure drop is:
Shim ReStackor gives you the capability to retune the stack to match this value and maintain the damping characteristics the manufacturer intended for your suspension. While these scaled stock damping rates may not be the optimum for the terrain and speeds that you ride it provides a solid starting point to begin tuning a new suspension.
In the end suspension performance all comes down to how the suspension performs on the "butt dyno". Nothing else really matters. After a couple of test rides the "butt dyno" may tell you the high speed rates are two stiff or the low speed rates are too soft. Shim ReStackor gives you the capability to run a few experiments with the clickers to determine the necessary high speed and low speed clicker settings to achieve the desired values and retune the stack to reduce the high speed damping and stiffen the low speed damping to match those clicker settings. The capability to do that in software and match specific clicker settings gives you the capability to fine tune the stack far beyond levels previously possible. This capability of ReStackor defines a new era in suspension tuning.
