CVD Damper Shock Absorber Bible

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ver. 2012/01/01 © Christopher Catto http://sites.google.com/site/cattovehicledynamics/Articles/damper-shock-absorber-guide

THE DAMPER / SHOCK ABSORBER BIBLE

Image: Tenneco Kinetic

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Table of Contents

To be populated Introduction This article covers material mostly extracted from four excellent books and data from the author’s racetrack work. Like other articles in this series, this one was initially created as notes on advanced topics in a way that was quickly accessible via mobile phone web connection for the author’s work at the track or factory.

The author dislikes the word “shock absorber”. However in the context of a spring and damper assembly, the term is in fact correct. Typically what absorbs a shock is a spring and what smooths out the resulting movement is the damper. The article is unashamedly written in SI units and for performance vehicles. Absorbing asperities – ¼ car spring rates Mediocre literature often describes the damper as a method of absorbing road irregularities. In fact this is the role of the spring! It compresses and then returns to its position. The problem is the masses supported on either side which like to oscillate at their natural frequencies. Here below are some ballpark figures for different vehicles. Sprung mass natural frequencies: Passenger car 6 to 12 rad/s Formula SAE 15 to 22 rad/s IRL car 30 to 45 rad/s F1 car 35 to 65 rad/s Unsprung mass natural frequencies: Passenger car 60 to 75 rad/s Formula SAE 90 to 120 rad/s IRL car 140 to 170 rad/s

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F1 car 150 to 200 rad/s Although the FSAE unsprung mass may be lower than that in an F1 car, its springs are also softer. For our initial calculations it may be useful to simplify a problem first and pick a good starting point, in order to later refine it and arrive to a more accurate solution. Hence we consider each wheel of the car individually (not linked) to choose a spring rate, and we assume a frequency at which we are happy for the car to “vibrate” while still being able to copy the road accurately or change direction willingly.

Basic Newtonian mechanics tells us that the stiffer the spring, the higher the natural frequency of the system. This is given by the relationship: Frequency rad/s = sqrt ( wheel rate / sprung mass) With rate in N/m, mass in kg So for the example of the rear spring of a Lotus Elise, with sprung mass 214kg and ride frequency of 10 rad/s, we would use a spring which is: Rate = 102 * 214 = 21400 N/m

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Most Eibach springs for example are rated in N/mm because the numbers are more easily represented in catalogues. This would correspond to 21.4 N/mm although the nearest available metric spring is likely to be a 20 N/mm.

In the case of a quarter car model where the spring/damper is in the vertical plane, we merely need to consider the unsprung mass on that portion of the vehicle and the motion ratio of the spring is 1:1 in such case. Later we will calculate motion ratios in a way that includes non-linearities and thus we can use the full equations. Hydraulic damper principles Before we go into the maths of the damper itself, we should consider that the damper has three main roles:

• absorb road irregularities • damp the sprung and unsprung masses at their resonant frequencies • control the rate of weight transfer during transients (braking, corner entry,

acceleration) There are primarily 3 types of damper. For the plus and minus points of each, the reader should consult any web resource on dampers.

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For high speed compression, almost all dampers use a shim stack to control the oil flow. This is a pyramid of very thin washers that bend when the oil is forced through the small holes under the stack that are otherwise sealed. A stack can be made in various shapes with different number of shims, different diameters and different thicknesses. The image shows the assembly for compression and rebound.

In compression, there may be a shaft orifice that is opened so that in most cases flow will flow this way and bypass the shim stack. The cross-section size of the orifice is often adjusted using an external dial.

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Many dampers instead have an adjustable needle valve that closes up an orifice.

In rebound, the oil simply flows behind the shim stack and pushes a large washer that is held in place by a small spring.

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Each manufacturer defines low, medium and high speed differently but on average the damping can be described as thus: Low speed (0 to 50 mm/s) is controlled by:

• Bleed holes / bleed discs • By-pass jet / adjustment needle • Leakage around piston

Medium speed (50-200 mm/s) is controlled by:

• Disc stack preload • Disc thickness

High speed (200+ mm/s) is controlled by:

• High speed portion of disc stack • Piston port size and fluid dynamics • Restriction of disc stack travel

Damping oscillation We can calculate the ideal damping ratio if we were to have no overshoot in the system. This is given by: Critical Damping constant = 2 * sqrt ( wheel rate * sprung mass ) Using our Lotus example again, we have: Cr = 2 * sqrt ( 21400 * 214 ) = 4280 Ns/m Which is sometimes expressed as 4.3 Ns/mm That is not to say that we are actually going to set up the dampers this way. In fact we may set them to be “softer”. The ratio between the real damping force and the Critical Damping force is called the damping ratio and is often expressed by the greek letter zeta. Damping ratio = C / Cr

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For race car applications it depends very much how the suspension is divided up between ride dampers, third dampers, anti-roll bars (have a small amount of damping due to the mountings) and not least the tyres whose side walls have a considerable effect that is often not calculated by most books! The zeta could be anywhere from 0.5 to 0.9 but Formula One cars are known to have higher values if they exploit the tall tyre side wall (FIA regulations restrict wheel diameter size). Keeping in contact with the road The purpose of the suspension is to keep the four precious tyre contact patches in touch with the asphalt. So the role of the damper is to allow the wheels to move as independently as possible, leaving the chassis undisturbed. This introduces the mathematical concept of transmissibility.

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If we observe that much of the road ondulation or vibration is imparted to the chassis, then the transmissibility is high. Transmissibility = amplitude of output / amplitude of input Different damping ratios have different transmissibility but the plotted curves show some interesting details: the peak is at the resonant frequency (the same as our ride frequency calculated earlier), and all lines intersect at a point which is 1.4142 * the resonant frequency. The lowest value of transmissibility will result in the best chassis control so we aim to achieve the best compromise. For racing applications what we want to achieve is minimum tyre load fluctuation. This is so that the tyres are always able to transmit lateral and longitudinal forces into the chassis.

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When looking at normalised RMS values for different damping, we can use the point at the curve’s trough. The example shows approximately 3000 Ns/m but this varies entirely depending on chassis and damper. Mass Spring Damper System equations If we consider the simple case of a mass attached to the ground via a spring and damper, we can describe the mathematics as follows:

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Hydraulic damper curves At low frequencies, we desire high damping in order to avoid being on the curves with a high peak. At higher frequencies, the requirement is the opposite. So in effect we would like to start off on one curve and then change to another curve. How can we achieve this? With the settings of the valves in the hydraulic damper. Such valves produce a damper curve that has a characteristic “knee” which allows them to achieve most of what is needed.

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To link our observations with the mathematics, we can propose these equations:

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We us a mean damping coefficient Cd which describes a linear relationship between damper force and shaft velocity. There is an asymmetry factor eD: literature from Milliken Milliken or practical studies from OptimumG suggests rebound damping may work well in practice if 3 to 4 times higher than compression damping. A progressivity factor represents the shape of the force-velocity curve, so for example lambda = 0 is a constant Coloumb friction, lambda = 1 is linear viscous damping and lambda = 2 is quadratic damping as could be achieved with the help of a Magneto-Rheologic fluid. Setting up dampers to be too stiff in rebound has the effect of “jacking down”. This is where the damper does not go back to its original position.

Simulation allows us to run sensitivity studies. For example on the offset value eD. If we plot normalised delta F with respect to eD, we can see where the curves are at their minimum. Again, this is for racing applications.

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Once we have the eD value we can then see the ratio between compression and rebound damping. When using Simulink or Modelica (Dymola/SimulationX), it is useful to create a step steer input and verify the response for different damping coefficients.

The goals are entirely dependent on the vehicle and its market. US vehicles for example tend to have a softer setup to European vehicles. In any case, the author feels a sharp handling car would not overshoot more than 18% and reach 90% steady state in 0.25 seconds.

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The velocity transition point needs to be prioritised depending on the use for the vehicle. For track racing purposes it may be of primary importance to tune the dampers for best turn-in response and as such the damper may be perfectly fine with a “knee” at 50 mm/s. A vehicle competing in the Paris Dakar may have a completely different set of results if damper speeds are far higher. When carrying out data logging there are two ways of measuring the vertical tyre loads: Pushrod load-cells � accurate, sold by companies such as Cosworth Electronics Math channels � approximation, but free Any datalogging software such as Pi Toolbox or Motec i2 can be used to set up a math channel to measure the force on the tyres:

Where MR = motion ratio “Car in Excel” & “Car in Simulink” Various of the equations found in this article are used within “Car in Excel” and “Car in Simulink” because it is easiest to practice with the formulae in the context of MS Excel or any tool the user wishes to use (like MATLAB for example).

When building the model in Simulink, it is necessary to define some assumptions: zs – z0 � difficult to measure consistently, hence zus – z0 � tyre deformation, noticeable in F1 but negligible with softer suspension

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and also some Design Requirements: desired overshoot < 5% settling time < 5s We apply Newton’s equations and integrate twice in order to obtain velocities and also displacements:

Road profile Mathematically we can run what the author describes as “statistics on road profiles”. The Power Spectral Density is one of these: imagine a value that tells on average how bumpy a road is.

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Damper velocity histograms Many data analysis softwares such as Pi Toolbox, with which the author is most familiar, can plot damper hystograms. These are are very useful tool to analyse how much time the damper spends within a given shaft speed. For example one can isolate cases for the vehicle travelling at high speed in a given part of the circuit: this allows tuning the dampers with a manageable set of information (not information overload) and then merging the information for different scenarios to find a suitable compromise.

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A good method to help set valving and damper clicks is to “gate” the track data based on longitudinal and lateral acceleration to tune for: Under positive longitudinal acceleration � tune dampers for squat Under negative longitudinal acceleration � tune dampers for pitch Under pure lateral acceleration � tune dampers for cornering Under high speed � tune dampers for track asperities and aerodynamics Damper rig Significant amounts of data can be obtained on a damper by running it on a dedicated rig such as the one depicted.

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The rig allows an input to be scheduled in such a way to realistically mimic track ondulations. Data logging and analysis can be done conveniently after each change in adjustment.

7-post rig A seven post rig is an invaluable tool for manufacturers and teams because it allows quick iterations and data analysis before taking a vehicle for real testing

The rig is typically actuated hydraulically and can produce various types of movement from sinusoidal to sawtooth to a full road profile.

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Magneto-rheological dampers [section to be expanded]

CVD Damper Shock Absorber Bible

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