Scaling of OEM Damping Rates
Spring-mass-damper theory defines two parameters controlling suspension performance: the suspension oscillation time constant (tau) and the damping coefficient (zeta). If two riders of different weights setup their bikes to have the same value of tau and zeta the suspensions will produce the same travel, feel, response and behavior when hitting the same bump.
The suspension oscillation time constant (tau) is defined as:
For a linked suspension system LK.tau (defined here) translates spring stiffness at the shock to the effective spring stiffness at the wheel. For a fork LK.tau is equal to one. Tau is defined as the ratio of mass (m) to spring rate (k) making tau essentially equivalent to race sag. Since riders select spring rates to match a specific race sag the value of tau for riders of different weights will be the same when race sag is matched. The coefficient g.c is a unit conversion relating mass to force and is a constant.
The second parameter (zeta) defines the suspension damping:
LR.fac (defined here) relates damping (c) and spring rate (k) at the shock to the forces actually delivered to the wheel in a linked suspension system. For a fork the value of LR.fac is equal to one. The damping coefficient (c) is simply defined as the shock absorber damping force divided by shaft velocity.
At high suspension speeds the shock only needs to damp the wheel. Since the unsprung wheel weight is the same when scaling a suspension setup, and the link ratio is the same, the only damping force adjustment needed is the correction for spring rate. Recognizing the un-sprung wheel weight is the same:
Low speed damping requires a correction for both weight and spring rate. Since LK.tau is the same for both setups that term cancels out. The equation for tau can be manipulated to determine the relationship of rider weight and spring rate:
"Stuffing" the tau relation into the zeta equation defines the change in damping coefficient (c) needed to correct low speed damping for a change in rider weight and spring rate:
Spring-mass-damper theory defines the need for a heaver rider to use stiffer damping. The increase in damping needed is defined by the ratio of spring rates.
ReStackor pro is able to directly compute shock absorber damping coefficients (c) and directly implement the above relationships for weight scaling a suspension. An example is here. The baseline Shim ReStackor code requires a couple more steps manipulating the above relationships to express the shock absorber damping coefficient in terms of shim stack stiffness.
Damping Requirements for a Heavier Rider
The damping coefficient is defined as:
The mass flow of oil through the suspension circuits is defined by the oil density, shock absorber shaft area and velocity. The ratio of the mass flow between the custom and stock configuration is:
Since the oil density and shaft area are the same when weight scaling a suspension the ratio of mass flow is directly proportional to the shaft speed.
Estimating the damping force requires a method to determine the pressure drop across the shock valve as a function of shim stack stiffness and the flow area. The Bernoulli equation provides that relationship.
In the above equation c.d is the discharge coefficient of 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 are also assumed to be the same for both bikes. With these values constant the pressure drop across the valve is a direct function of the shim stack deflection flow area.
The above expression defines the relationship of shim stack stiffness and shock absorber damping force. Stiffer shim stacks deflect less, produce less flow area and result in a higher pressure drop across the valve. The force from this pressure drop acts directly on the shock absorber damping rod producing the damping force of the shock.
The direct relationship between pressure drop and damping force allows the damping force to be directly substituted when used as a ratio in weight scaling the suspension. The shock absorber damping coefficient (c) defines the relationship between damping force and shaft velocity. Substituting the damping coefficient for the value of damping force gives:
There are two ways to compare the damping force of a shock. One way is to compare the damping force at a fixed shaft velocity. The other way is to compare the difference in shaft velocity needed to produce the same damping force.
Comparing damping force at the same shaft velocity gives the expect result of the flow area being proportional to the square root of damping force and pressure drop. Comparing shock absorbers on the basis of the shaft velocity required to produce the same damping force gives a much different result:
At the same damping force the stack face flow area is directly proportional to the damping coefficient.
Shim ReStackor compares the deflection of two stacks on the basis of the same force applied to the shim stack face. This is analogous to case 2 where the difference in shaft velocity is determined to produce the same damping force.
For weight scaling the Shim ReStackor exercise is to determine the shim stack modifications necessary to produce a stack face flow area that is proportional to the change in damping coefficient desired for weight scaling the suspension. For low speed damping the change needed is directly proportional to the spring rate change. For high speed damping the constant unsprung wheel mass results in a flow area change requirement that is proportional to the square root of the spring rate change.
It is that simple.
For an example application of this process check out the crf450r example in the sample applications section of the ReStackor web site.
Shim ReStackor Weight Scaling Spreadsheet
Weight scaling a stock shim stack follows these simple steps:
It is that simple.
Weight Scaling Damping Rates of the Stock Suspension for a Rider of Different Weight
The fundamentals of spring-mass-damper theory define two parameters that control suspension response. Tau defines the oscillation time constant of the suspension and will be the same for two riders of different weight if the spring rates have been set to produce the same race sag. Zeta defines the damping characteristics of a suspension. Suspension link ratio effects are determined by the parameters LK.tau and LR.fac as defined here. Maintaining the suspension response, feel and performance requires the shim stack stiffness to be modified in proportion to the spring stiffness:
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 the stock suspension, and the rider weight it was designed for, to your weight and spring rate. Application of the fundamentals of spring-mass-damper theory give you the capability to restore the suspension response, feel and behavior the manufacturer intended for your bike. While these scaled damping rates may not be the optimum for the terrain and speeds that you ride it will produce a solid initial suspension setup to begin the tuning process.