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Spring-Mass-Damper Equations
Force balance describes oscillation behavior
Control of oscillations in a spring-mass-damper system is a classic engineering control systems problem and appears within the first couple of chapters of many standard texts on control system design. The response of the spring-mass-damper system used in a motorcycle suspension is a well studied problem. The system consists of the cart below with weight (m), a spring with stiffness (k) with the oscillations damped by 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 shock absorber damping coefficient (c) is defined as force produced by the shock divided by the shaft velocity:
Combining these three elements into a force balance describes the cart motion:
This equation completely describes the cart motion through the entire cycle of the oscillatory 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 a difficult equation to solve. Texts on control system theory thoroughly work through the solution process so we are not going to muddle through that here. Instead, we jump directly to the solution. The solution is written in terms of the systems oscillation time constant t (tau) and a damping coefficient z (zeta):
If the value of the damping coefficient z is less than one the solution for a step response is:
and for damping coefficients z greater than one:
The response of the system to an impulse blow is defined by:
It is important to understand that 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 description of the motion of a spring-mass-damper system in the same sense that F=ma is an exact relationship between force and acceleration.
The form of the solution gives some insight into how the suspension will behave. For zeta < 1 the sin term shows that the behavior will be periodic and oscillate with a frequency controlled by the value of tau. Stiffer springs will make the system oscillate at a higher frequency. Increased weight will make the system oscillate at a lower frequency. If the spring rate and weight are both changed so that the value of tau remains constant there will be no effect. Within the sin term zeta also effects the oscillation period, but being buried inside the square root and tangent term the effect of zeta is weaker on the oscillation frequency.
The zeta < 1 solution also includes an exponential term involving zeta, tau and time. The negative sign in the exponential term tells you it is a decay term. At infinite time the entire right hand side of the equation will be forced to zero. The rate of decay is a function of both tau and zeta. High zeta values accelerate the decay rate. That behavior is reasonable since zeta depends on the damping force coefficient c and high damping rates are intuitively expected to reduce oscillations of the system.
Suspension Response to a Step-Up
The response of a spring-mass-damper system to a sudden 10" step-up is plotted below. Ideally the wheel would suddenly jump to the top of the 10" step and stay there without oscillations. But, because the wheel has mass, it takes some time to accelerate the wheel in the vertical direction and once moving the momentum of the wheel causes the wheel to overshoot the top of the step. Plotting the above equation shows this behavior. The relationships of spring-mass-damper theory give you the capability to not only determine that the wheel will overshoot the top of the step but also determine the magnitude of the overshoot.
Suspension response as a function of time after a 10 inch step. Wheel is slow to respond to step for the critically damped case.
Under damped suspension response zeta<1:
If the suspension is under damped (z< 1) the suspension will oscillate and baby buggy a couple of times before settling down on top of the step. While this is an intuitive result it is important to note that the equations duplicate this behavior. In addition the equations give you an exact numerical description of the process. For a damping coefficient of 0.2 (z=0.2) the suspension will overshoot the 10" bump by about 5", then undershoot by 3" and baby buggy a couple more times before setting down on top of the step. All of this motion occurs within 1.2 seconds of hitting the bump. As the value of the damping coefficient (z) increases the overshoot decreases as well as the number of baby buggy oscillations and the time required to damp the system.
These relationships can be directly applied to your suspension if you know the value of the shock absorber damping coefficient (c). Calculation of the damping force produced by your suspension as a function of the shim stack stiffness, oil viscosity and valve port configuration is the central application of ReStackor.
Over damped suspension response zeta>1:
For zeta equal to one (z=1) the suspension is critically damped. Critically damped means the suspension motion asymptotically approaches the top of the 10" step without any overshoot. With no overshoot there is no baby bugging . Increasing the damping coefficient above 1.0 results in a progressively slower response i.e. it takes longer for the suspension to respond to the 10" bump. For the critically damped case the suspension responds to the 10" step in about 0.8 seconds. Increasing the damping coefficient to 1.6 requires more than 1.2 seconds for the suspension to reach the 10" mark due to the heavily damped motion of the suspension.
Initially, you might expect you would want your suspension to be critically damped with no baby bugging. If this is your thinking you need to take a closer look at the above plot. The critically damped case requires 0.8 seconds to respond to the 10" bump. An under damped case with a damping coefficient of 0.4 allows the suspension to respond to the 10" 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. The reduced damping causes the suspension to baby buggy a couple of times after the step, but as shown in the plot above these oscillations damp within a fraction of a second. So the trick to suspension tuning is to find and under damped case that is soft enough to allow the wheels to float over bumps, yet stiff enough to quickly damp oscillations. That balance is totally a case of personal preference and a strong function of the terrain you ride.
Picking a damping rate is further complicated by independently adjustable compression and rebound. You could be under damped on compression and over damped on rebound, or under damped on both. Finding the sweet spot with adequate compression damping to keep the suspension from bottoming and rebound damping that does not pack the suspension is something you have to experiment with.
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 suspensions around the phantom weight of the stock factory rider. If you change your bikes spring rate to get the sag and suspension geometry correct you are going to have to change the suspensions damping rate to restore the suspension feel and response the manufacture intended for the spring rates of the stock suspension.
Spring-mass-damper theory gives you the relationships needed to correct damping rates for a spring rate change. To implement the process you need to accept the following assumptions:
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After winning several national championships the manufacture knows something about setting up MX and road racing motorcycle suspensions.
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Some of that suspension technology makes it from the track to production bikes.
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Suspensions on stock production bikes are built around the weight and spring rate of the phantom stock factory rider. The weight the suspension was designed for can be determined from the spring rate used on the stock bike.
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The fundamentals of spring-mass-damper theory can be extracted from the ideal world of engineering theory and applied to solve practical real world problems.
From spring-mass-damper theory there are two fundamental parameters that control the feel and characteristics of a suspension. These are the suspension oscillation time constant (tau) and the system damping coefficient (zeta). If two bikes of different weights, with different dampers and a different spring rates are setup so that tau and zeta match then the two bikes will have exactly 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 the square root of race sag. The coefficient g.c is a unit conversion that relates the units of force and mass, g.c is a constant. If you have set your spring rate to match the factory recommended race sag then the value of the suspension 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. Calculation of the suspension damping coefficient (c) as a function of shim stack stiffness, oil viscosity and valve port configuration is the central focus of ReStackor.
Correcting damping rates for a change in rider weight
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ReStackor pro: The capability of ReStackor pro to evaluate the effect of suspension valve geometry on damping rates gives you the capability to scale the damping rates of your old bike and the years of suspension tuning that went into development of the shim stack on that bike to your new bike. This will setup your new bike with the same suspension feel and response. While that may not ultimately be the best damping configuration for the new bike it will setup the suspension with a familiar feel and response and provide a solid baseline for you to begin the tuning process on the new bike (more).
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Shim ReStackor: The fundamentals of spring-mass-damper theory give you the relationships needed to scale the damping rates the manufacture developed for the phantom weight of the stock factory rider and scale those damping rates to correct for a change in spring rate and rider weight. This restores the suspension response, feel and behavior the manufacture intended your suspension to have and setups the bike to match the spring rates required for you weight (more).
Weight Scaling Stock Suspension Damping Rates
From the fundamentals of spring-mass-damper theory there are two parameters that 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 and relates the effect of weight and spring rate to suspension response. To maintain the same suspension feel, response and behavior a change in spring rate requires a corresponding change in damping rate to keep the value of zeta the same.
ReStackor gives you the capability to define the shim stack changes needed to adjust the value of zeta. This gives you the capability to scale damping rates from one bike to another or correct the damping rates of the stock suspension for changes in rider weight and spring rate.
In the end performance of a suspension all comes down to how the suspension performs 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 on the "butt dyno" gives you the capability to quantify the specific changes needed and ReStackor gives you the capability to relate those clicker changes to specific high speed or low speed shim stack changes to match those specific clicker settings. The capability to do that in software, match specific clicker settings and fine tune the high speed and low speed shim stack configuration gives you the capability to fine tune your suspension far beyond the limits previously possible. ReStackor defines a new era in suspension tuning.
