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Spring-Mass-Damper Systems

Suspension Tuning Basics

Controlling oscillations in a spring-mass-damper system is a well studied problem appearing in a large number of engineering text books. Two fundamental parameters, tau and zeta, control the frequency and amplitude of oscillations and those parameters are simple functions of mass, spring rate and damping.  

If two bikes of different weights, with different rider weights and different spring rates are setup so that the values of tau and zeta match then the suspension response of those bikes will retrace the values shown in the plot.

Tuning suspensions in terms of tau and zeta allows the suspension characteristics of one bike to be scaled to another and corrections in damping rates to compensate for changes in spring rate or rider weight. 

Spring-Mass-Damper System

The spring-mass-damper system consists of the cart below with weight (m), a spring with stiffness (k) and a shock absorber with a damping coefficient of (c).

The equations describing the cart motion are derived from F=ma. The force required to accelerate the cart mass is:

and the force produced by the spring:

The damping force of the shock absorber is described by the damping coefficient (c) simply defined as force produced by the shock divided by the shaft velocity:

Combining these elements into a force balance describes the cart motion:

The above equation describes the oscillatory motion of the cart and includes effects of inertia as the cart is accelerated and decelerated, changes in spring force over the stroke and changes in damping force with shaft velocity throughout the entire cycle of motion.

Solution to the Equation of Motion for a Spring-Mass-Damper System

The force balance for the cart is a second order partial differential equation with constant coefficients. The fact that the equation has a name is a clue that it is difficult to solve. Texts on control system theory thoroughly work thoroughly the mathematics of the solution process so we will not muddle through that here. Instead, we jump directly to the solution written in terms of the oscillation time constant tau and a damping coefficient zeta:

If zeta is less than one (the usual case for suspensions) the cart motion is described by:

If zeta is greater than one the equation becomes:

The response of the system to an impulse blow is defined by:

It is important to understand the above equations are not an approximation or estimate that sort-of follows the behavior of a spring-mass-damper system. The above equations are exact and provide an exact mathematical relationship describing the oscillation motions of a spring-mass-damper system in the same sense that F=ma is an exact relationship between force and acceleration.

Inspection of the equations give some insight into how a suspension will behave and the influence of tau and zeta on the performance of a suspension. The impulse response equation for zeta < 1 has the following behavior:

  • The sin term shows the response will be periodic with an oscillation frequency controlled by the value of tau.

  • The value of tau is defined by the ratio of mass to spring rate.

    • Stiffer springs will increase the oscillation frequency.

    • Increased weight will decrease the oscillation frequency.

    • If the ratio of mass to spring rate is the same the value of tau will be the same and the oscillation frequency will be the same, regardless of weight or spring rate.

  • The oscillation frequency is also influenced by the value of zeta, but with the zeta term buried inside the square root the influence of zeta on oscillation frequency is weak.

  • The exponential term describes the rate of decay. 

    • At infinite time the negative term in the exponential forces the entire right hand side of the equation to go to zero indicating the suspension will return to its initial ride height after the oscillations die out.

    • The zeta term in in the exponential controls the rate of decay. Increasing the damping coefficient increases the value of zeta making the decay faster. 

The above behaviors are consistent with what you would intuitively expect from a suspension. The difference, and the reason for developing the equation, is the equation can be used to quantify the suspension response, determine the effect of a spring rate or damping force change, or the effect of multiple simultaneous changes in spring rate, mass and damping to see how the suspension responds. The more direct use is to determine the change in damping needed to keep the suspension response the same when the rider weight and spring rate are changed. 

Suspension Response to a Step-Up

Suspension response for a sudden 10" step-up computed using the above equations is plotted below. Ideally the wheel would suddenly jump to the top of the 10" step and stay there with no oscillations. Because the wheel has mass it takes some time to vertically accelerate the wheel. Once moving, the momentum of the wheel causes the suspension to overshoot the top of the step. The combination of overshoot and wheel inertia setup the oscillations show below. 

Motorcycle suspension theory of suspension response accounts for the combined effect of weight, spring rate and damping on the oscillation characteristics of the suspension.

Suspension response as a function of time over a 10 inch step. Stiff damping slows the wheel response.

Under damped suspension response zeta<1:

If the suspension is under damped (zeta< 1) the suspension will oscillate and baby buggy a couple of times before settling down on top of the step. For a damping coefficient of 0.2 (zeta=0.2) the suspension will overshoot the 10" bump by about 5", then undershoot by 3" and baby buggy a couple more times before settling down on top of the step. All of this motion occurs within 1.2 seconds of hitting the bump. 

Increasing the damping coefficient (zeta) decreases the overshoot as well as the oscillations. The down side is a heavily damped suspension takes longer to respond to a bump. The above plot shows higher zeta values take longer to clear the top of the step. For the critically damped case (zeta=1) the suspension requires 0.8 seconds to respond to the 10" step. An under damped case with a damping coefficient of 0.4 responds to the bump in 0.2 seconds, about 4 times faster. Four times faster is a big deal. The faster response allows the wheel to float over the bump, aka plushness. If the wheel can not clear the bump then the bump pounds directly into the bars. 

Suspension Performance Basics

Spring-Mass-Damper theory defines suspension performance in terms of two parameters, tau and zeta.

Those parameters define the effect of mass, spring rate and damping on suspension performance. As long as the values of tau and zeta are the same the performance, response, feel and behavior of the suspension will be the same regardless of rider weight or spring rate. Preserving suspension performance when the rider weight is changed requires changing both the spring rate and damping to keep the values of tau and zeta constant. 

Weight Scaling a Stock Suspension

Modern motorcycles use sophisticated suspensions with independently adjustable compression, rebound, high speed and low speed damping. While the suspensions are sophisticated, mass production forces manufactures to build the suspension around the weight of some phantom stock factory rider. Changing the spring rate in your bike to get the sag right drives the suspension into an under-damped or over-damped condition directly impacting performance.

Restoring performance to the level the factory intended for your bike requires modifying the damping rate to match the spring rate and restore the factory value of zeta. That is the basic premise of weight scaling based on the following assumptions:

  • After winning several national championships and GP titles manufactures know something about the setup of MX and road race suspensions.

  • Some of that suspension tuning knowledge makes it from the track to production bikes.

  • Weight - spring rate - and damping are intimately coupled parameters that control suspension performance. The weight the suspension was designed for can be determined from the spring rate used in the stock suspension.

  • The fundamentals of spring-mass-damper physics can be extracted from the ideal world of engineering theory to solve practical real world problems.

Spring-mass-damper theory defines two parameters tau and zeta that control suspension response. If two bikes of different weight, with different dampers and different spring rates are setup so that the values of tau and zeta match then the suspension wheel travel and response on any given bump will be exactly the same giving the two suspensions the same response, feel and behavior. 

The suspension time constant (tau) is defined in terms of mass and spring rate. For a linked rear suspension the actual spring rate at the wheel is modified by the travel ratio (TR) and link ratio (LR) of the linkage creating the parameter LK.tau defined here. For a fork LK.tau is equal to 1.0. 

Defining tau as the square root of the mass (m) divided by spring rate (k) makes tau essentially equivalent to the square root of race sag. The unit conversion (g.c) relates the units of force to mass, it's a constant. If you have reset your spring rate to match the factory recommended race sag then the value of tau is already matched to the stock factory setup.

The damping rate (zeta) is defined in spring-mass-damper theory as:

Here the added parameter is the shock absorber damping coefficient (c) simply defined as the force produced by the shock divided by the shaft velocity and the effect of link ratio summed up by the LR.fac parameter defined here. Changing the suspension spring constant (k), the rider weight (m) or link ratios (TR, LR) requires the damping coefficient (c) to be modified to keep the value of zeta constant. If your setup has a higher spring rate to compensate for a heavier rider but you are still using the stock damping coefficient (c) then the value of zeta has been reduced driving the suspension into an under damped condition. A lighter rider with a lighter spring rate would be over damped. 

The fix is simple. Change the damping coefficient to keep zeta the same.

Frequency, amplitude and damping of any spring-mass-damper system will be identical, regardless of weight or spring rate, as long as the values of tau and zeta match. Correcting the value of zeta after a mass or spring rate change requires damping to be modified to restore the initial value of zeta. 

Correcting damping rates for a change in rider weight

Weight scaling of damping rates is a straight forward process. The problem is nobody can tell you the value of the stock damping coefficient (c) nor the shim stack modifications needed to correct for a change in mass or spring rate. ReStackor gives you that capability. 

  • ReStackor pro: ReStackor pro gives you the capability to scale suspension setups from one bike to another and account for changes in suspension valve geometry, fluid viscosity and bleed circuit configurations. This gives you the capability to scale the factory suspension setup for a change in weight or some custom suspension setup from the valve geometry of your old bike to your new bike with a different valve geometry.  (more).


  • Shim ReStackor: The fundamentals of spring-mass-damper theory define the relationships needed to scale damping rates the manufacture developed for the stock factory rider to the weight and spring rate of your suspension setup based on shim stack stiffness. (more)

Weight Scaling Stock Suspension Damping Rates 

The fundamentals of spring-mass-damper theory define two parameters that control suspension response. Tau defines the suspension oscillation time constant and will be the same for two riders of different weights if the suspensions are setup to have the same race sag. Zeta defines the damping characteristics and relates the effect of weight and spring rate to suspension response. Maintaining the suspension response, feel and behavior after changing the spring rate requires the damping rate to be modified to restore the factory value of zeta. 

ReStackor gives you the capability to define the shim stack modifications needed to restore the initial value of zeta. This gives you the capability to scale damping rates from one bike to another or correct damping rates for changes in spring rate and rider weight.

In the end, performance comes down to how the suspension feels on the "butt dyno". After a couple of rides the "butt dyno" may tell you the high speed rates are too stiff or the low speed rates are too soft. Experimenting with the clickers gives you the ability to quantify specific changes in damping needed to correct the suspension and ReStackor gives you the capability to relate those clicker changes to specific shim stack modifications needed to match those clicker settings. The capability to define the shim stack in software, match specific clicker settings and fine tune high speed and low speed damping gives you the ability to tune your suspension far beyond the limits previously possible. ReStackor defines a new era in suspension tuning.