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 text books on engineering control system theory. The frequency and amplitude of oscillations are determined by two fundamental parameters, tau and zeta, which are 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 will retrace the values shown in the plot.
Tuning suspensions in terms of the values of tau and zeta allow the suspension characteristics of one bike to be scaled to another and damping rates to be corrected for changes in spring rate.
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 including the effects of inertia and damping force through 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 the damping coefficient zeta is less than one the cart motion is described by:
and for damping coefficients zeta greater than one:
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 follow 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 gives some insight into how a suspension will behave and the influence of tau and zeta on the performance of a suspension. The impulse response for zeta < 1 shows the following behavior:
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The sin term shows the response will be periodic with an oscillation frequency controlled by the value of tau.
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The value of tau is defined by the ratio of mass to spring rate.
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Stiffer springs will increase the oscillation frequency.
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Increased weight will decrease the oscillation frequency.
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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.
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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.
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The exponential term describes the rate of decay.
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At infinite time the negative term in the exponential forces the entire right hand side of the equation to zero indicating the suspension will return to its initial ride height after the oscillations die out.
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The zeta term in in the exponential controls the rate of decay. Increasing the damping coefficient increases the value of zeta and the rate of oscillation decay.
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The above behaviors are all consistent with what you might intuitively expect for the behavior of a suspension. The difference is the equations can determine the effect of multiple simultaneous changes in spring rate, mass and damping and determining the change in damping coefficient needed to keep the suspension response the same. That is more than you could expect form even a well trained intuition.
Suspension Response to a Step-Up
The response of a spring-mass-damper system to 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 wheels have 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.
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. 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. Lower damping causes the suspension to baby buggy a couple of times after the step, but as shown in the plot, the oscillations damp within a fraction of a second.
Suspension Performance
Spring-Mass-Damper theory defines suspension performance in terms of two parameters, tau and zeta.
These parameters define the relationship of mass, spring rate and damping on the performance of a suspension. 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 the suspension performance when the rider weight is changed requires both the spring rate and damping to be changed 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 the phantom stock factory rider. Changing your bikes spring rate to get the sag right drives the suspension into an under-damped or over-damped value of zeta directly impacting performance.
Restoring performance to the level the factory intended for your bike requires modifying the damping rates to restore the factory values of zeta. Weight scaling of damping rates is based on the following assumptions:
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After winning several national championships and GP titles the manufacture knows something about the setup of MX and road race suspensions.
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Some of that suspension tuning knowledge makes it from the track to production bikes.
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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.
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The fundamentals of spring-mass-damper theory 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 the response, feel and behavoir of a suspension. If two bikes of different weight, with different dampers and different spring rates are setup so that the values of tau and zeta match the suspension wheel travel and response on any given bump will be the same and the two suspensions will have the same response, feel and behavior.
Spring-Mass-Damper theory defines the suspension oscillation time constant (tau) as:
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, and is a constant. If you have reset your spring rate to match the factory recommended race sag then the value of tau will inherently match the stock factory setup.
The second parameter from spring-mass-damper theory (zeta) defines the damping performance of the suspension:
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. Changing the suspension spring constant (k) or the rider weight (m) requires the damping coefficient (c) to be modified to keep the value of zeta constant. If the damping coefficient is not changed the suspension is driven into an under-damped or over-damped condition.
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.
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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 either scale the factory stock suspension setup or the custom suspension setup from your old bike to your new bike. (more).
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Shim ReStackor: The fundamentals of spring-mass-damper theory specify the relationships needed to scale the damping rates the manufacture developed for the stock factory rider to the weight and spring rate of your suspension setup. Weight scaling restores the suspension response, feel and performance the manufacture intended your suspension to have (more).
Weight Scaling Stock Suspension Damping Rates
The fundamentals of spring-mass-damper theory define two parameters that control suspension performance. Tau defines the suspension oscillation time constant and will be the same for two riders of different weight if the suspensions are setup with the same race sag. Zeta defines the damping characteristics and relates the effect of weight and spring rate to the 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 changes 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.
