Theory

jump to our sub-topics:
Weight Transfer
Anti-Squat
Graphical Method
How Much Anti-Squat
Anti-Squat Curve
Wheel Rate
Anti-Rise

Introduction

There are four key factors to consider when designing the kinematics of a bike suspension system:
- Wheel path
- Wheel rate
- Braking response
- Acceleration response

Each of these factors affects the overall behaviour of the bike. Consequently, it would be advantageous if we could adjust each of these factors independently, or at the very least, lessen the compromise between these factors.

Wheel path determines the dynamic geometry of the bike, meaning the way the bike ‘changes shape’ as the suspension moves. For example, a bike with a near vertical wheel path will have a reasonably constant chainstay length, while a bike with a rearward wheel path will have a varying chainstay length. As is well understood, the chainstay length has a significant effect on the handling of the bike.
Wheel path also determines how well the wheel can react to forces from different directions. For example, forces due to rider inputs (cornering, pumping etc) are generally vertical in direction, while forces due to terrain (especially rough terrain) are more rearward in direction.

Wheel rate is dependent on the wheel path and the shock actuation mechanism. For the majority of bike suspension systems, the wheel rate can be easily tuned by changing the actuation of the shock. This provides a means for independently altering the wheel rate, without affecting the other key factors.

Braking response is dependent on the rotation of the structure that the brake caliper is attached to. With the majority of suspension systems, the brake caliper is attached to the member that defines the wheel path (aka wheel carrier member, or rear triangle). Therefore, with these systems, braking response cannot be designed independently of wheel path. A compromise must be made.
Some suspension systems (floating caliper, split-pivot, Trek ABP) have the brake caliper mounted on a separate member, allowing the braking characteristics to be designed independently of the other key factors.

Acceleration response is dependent on the wheel path and the orientation of the drivetrain.
The bottom bracket location is usually solely determined by ergonomics. i.e. it is located to provide a specific desired geometry. It is not beneficial to move the bottom bracket as a means for tuning the acceleration response. Therefore, we can consider that the acceleration response cannot be designed independently of wheel path. A compromise must be made.
Some suspension systems feature an idler mounted on the front triangle, as a means for ‘correcting’ the chain line after designing a rearward axle path. By varying the location of the idler, it provides a means of adjusting the acceleration response, allowing it designed independently of wheel path.
One limitation of this method is that the idler is fixed at a single location on the front triangle. This ultimately limits the ‘tunability’ of the acceleration response, meaning that the ‘fine tuning’ can still only be achieved by adjusting the wheel path.

With all of these key factors being dependent on wheel path, it is inevitable that some (or all) of them are compromised when designing a suspension system to be good all-round performer.
It would be advantageous to provide a suspension system that can facilitate less compromise between these key factors.

i-track provides a suspension system where there is complete freedom to adjust the mechanism that controls the path of the idler. With this system, the acceleration response can be tuned with greater flexibility than is possible with any other suspension system on the market.

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Weight Transfer

When a bike and rider accelerate forwards, a phenomenon called weight transfer occurs. This means that under powered acceleration, the rear wheel carries more weight (and the front wheel carries less weight) than if the bike was rolling at constant speed.
If the bike has rear suspension, then the extra weight carried by the rear wheel (due to weight transfer) is usually exhibited by the suspension system compressing.

Weight transfer is unavoidable during acceleration, and occurs equally with all suspended and non-suspended vehicles undergoing constant acceleration.
The presence of suspension does not alter the amount of weight transfer, but it can affect the timing of weight transfer.

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Anti-Squat

With current suspension technology, it is possible to design the suspension system in such a way that chain tension (from pedalling forces) causes an extension force in the suspension system. This extension force can counteract the compression that would otherwise occur under weight transfer. This phenomenon is known as ‘Anti-Squat’.

In order to objectively compare the pedalling characteristics of different suspension designs, it is necessary to quantify the amount of anti-squat exhibited by each suspension system. We can do this by recognising two following important behaviours:

  1. The extension force caused by chain tension perfectly balances the compression force caused by weight transfer. The suspension system doesn’t extend or compress under pedalling. We can define this particular behaviour as 100% Anti-Squat.
  2. The chain tension does not cause any extension or compression force in the suspension. The suspension system will compress under acceleration, due to weight transfer alone. We can define this particular behaviour as 0% Anti-Squat.

Using these definitions, we can assign a % value of Anti-Squat to any suspension system, to describe the way it behaves under powered acceleration.
In order to calculate the % Anti-Squat for a particular suspension design, it is necessary to perform a graphical analysis on the suspension/drivetrain system.

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Graphical Method for Quantifying Anti-Squat


Most suspension bikes available today have the bottom bracket mounted on the chassis (aka front triangle). Most suspension bikes also have a chain drivetrain, transmitting power directly from a front sprocket (rotating at the bottom bracket) to a rear sprocket on the rear wheel.
For these suspension systems, there is a reasonably well-known graphical method that suspension designers can use to calculate the % Anti-Squat. This method is described in Tony Foale’s book ‘Motorcycle Handling and Chassis Design: the art and science’, and also in Dave Weagle’s patent (US7128329).

This graphical method involves drawing a series of construction lines through specific suspension and drivetrain points, and noting their intersections. The end result gives a number that can be converted into a % Anti-Squat for that suspension system.
The % Anti-Squat describes how a particular suspension system will behave, in relation to the previous definitions of 0% and 100% Anti-Squat.

For suspension designs where the bottom bracket is mounted on the front triangle, with a ‘conventional’ drivetrain, the graphical method is as follows:

  1. Draw a line from the rear axle to the instant centre of the wheel carrier member relative to the front triangle (for a single pivot, this instant centre is the main pivot. For a 4-bar linkage, it is the intersection of lines through the two intermediate links).
  2. Draw a line through the upper chain line.
  3. Note the intersection of lines 1 and 2.
  4. Draw a line from the rear wheel contact patch, through point 3, and extend it over the front axle.
  5. Draw a vertical line through the front axle.
  6. Note the intersection point of line 4 and line 5, and compare it to the height of the centre of gravity, and the height of the rear wheel contact patch.

If the point obtained in step 6 is at the same height as the centre of gravity of the suspended mass, then the bike exhibits 100% anti-squat at that point in travel.
If the point obtained in step 6 is at the same height as the rear wheel contact patch, then the bike exhibits 0% anti-squat at that point in travel.
For any points between/outside of this range, the % Anti-Squat can be calculated by interpolation/extrapolation.

Note that the height of the centre of gravity of the suspended mass (which is mostly made up by the rider) plays a significant role in the calculation of % Anti-Squat.
Also note that the longitudinal position of the centre of gravity plays no role in the calculation of % Anti-Squat.

This method is also applicable to suspension systems which feature an idler mounted on the front triangle. The only difference being that the line drawn in step 2 should be the chainline from the rear sprocket to the idler.


There are a number of suspension designs where this graphical method does not apply. These include:
- Unified Rear Triangle (Trek, Klein)
- Independent Drivetrain (GT i-Drive, Mongoose, Lapierre)
- Swingarm Mounted Idler (Corsair, Balfa, Empire, Commencal)
- i-Track Suspension

The engineers at i-track suspension have a strong understanding of vehicle dynamics, and have been able to devise new graphical methods, to enable the % Anti-Squat to be accurately quantified for all of these suspension systems. These methods are currently kept as a ‘trade secret’.

These graphical methods serve as a fundamental tool for suspension design, allowing the designer to predict the pedalling behaviour of any proposed suspension design.

ThH

How Much Anti-Squat?

Traction vs. Stability

There is no ‘perfect’ amount of Anti-Squat; it very much comes down to riding style, terrain, and rider preferences. Some bike manufacturers like their bikes to have a particular feel when pedalling. This might be based on whether the bike is proven to be faster, more efficient, or simply more enjoyable to ride.

Firstly, the designer needs to choose whether the bike is being designed for traction or for stability during acceleration.
‘Traction’ is a well understood term, so there’s no need to explain what that means.

‘Stability’ is the term we use to describe a suspension system’s ability to stay near its equilibrium position while transmitting power through the rear wheel. A stable system will also transmit power more efficiently. 

As we know, when dealing with tyre pressure, a lower pressure provides more grip, however tyre deformation under cornering loads result in sloppy handling.  The same principles are applicable to suspension compliance. 
With softer springs, the suspension is more compliant. However, having soft springs causes a multitude of vehicle handling issues, such as excessive chassis movement shifting the vehicle away from its ‘ideal’ equilibrium geometry.

When designing suspension for bikes to perform well under acceleration, the same trade-off exists. Do we design the suspension for traction or for stability?
To understand the answer to this, it’s necessary to know what the bike’s intended purpose is, and whether an increase in traction or stability will improve the bike’s overall performance.

If a vehicle has plenty of power, but not much traction, then the acceleration of that vehicle will be limited by the amount of traction. The vehicle is said to be ‘traction limited’. A traction limited vehicle will benefit from an increase in traction.

If a vehicle has plenty of traction, but not much power, then the acceleration of that vehicle will be limited by how much power can be transmitted into the terrain. The vehicle is said to be ‘power limited’. A power limited vehicle will benefit from an increase in power (or an increase in the efficiency of power delivery, i.e. an increase in stability).

We’re of the opinion that when mountain biking, the situation of being traction limited under acceleration usually only occurs on a steep pinch climb, and only makes up for very small portion of overall riding.
Therefore it makes sense to optimise power delivery and stability, which can be achieved by carefully tuning the anti-squat to minimise ‘pedal bob’.

Designing for Stability:

100% Anti-Squat is the theoretical value required to prevent suspension compression due to weight transfer. As we have discussed, weight transfer is associated with longitudinal accelerations.
So if there were no vertical forces acting on the suspension, we could simply design for 100% AS, and the system would be relatively stable during acceleration.

However, in reality, there are significant vertical accelerations affecting the system in conjunction with the longitudinal accelerations. Specifically, vertical oscillations of the rider’s centre of gravity, caused by the biomechanics associated with pedalling, have a significant influence on suspension behaviour. This oscillation is caused by a combination of upper body movement and leg movement.

Even if the rider was to maintain a perfectly still upper body (which is not very efficient!), the motion of legs turning the cranks still creates a vertical imbalance. If the rider stands up while pedalling, then the amplitude of this oscillation is much greater than if sitting.

This is one of the primary reasons that suspension design for bicycles is more challenging than for motorcycles, and why bicycles can benefit from more sophisticated suspension systems.

As it turns out, when a rider exerts their peak crank torque (usually when the crank passes through the 4-6 o’clock position, when viewed from the right-hard side), their centre of gravity is accelerating upwards, which exerts an additional downward force on the bike chassis. Therefore, a bike with 100% anti-squat will still compress when pedalling due to this additional downward force on the chassis.

This phenomenon can be addressed by using more than 100% Anti-Squat. How much more depends on how much (and the timing of) CoG movement the rider, but somewhere between 100% and 150% is usually a good start.

Designing for Traction:

The action of the bike squatting under powered acceleration can be considered as a useful mode of suspension movement.
If a bike is designed to be relatively stable under acceleration (let’s say around 120% AS), then the suspension won’t compress much during each pedal stroke. The tyre force at the rear wheel will vary according to how much weight transfer is occurring, which is proportional to the pedalling force. Pedalling force is usually uneven, especially when pedalling out of the saddle. So, during the power zone of each pedal stroke, the rear wheel will have an increased tyre force, while in between pedal strokes the tyre force is less (closer to static levels).

If the bike has more than 120% Anti-Squat, then the suspension will extend (centre of gravity rises) during each pedal stroke. This upward movement of the centre of gravity causes a temporary increase in tyre force at the contact patch, additional to the increase in tyre force that occurs due to weight transfer.
In between pedal strokes, there is less weight transfer and tyre force reduces. At the same time, the suspension returns to its normal position (the centre of gravity falls), and the tyre force further reduced.
An uneven pedalling action produces a widely varying tyre force, causing an overall reduction in traction. 

If a bike has less than 120% Anti-Squat, the suspension will compress (centre of gravity will lower) during each pedal stroke. This downward movement of the centre of gravity causes a temporary decrease in tyre force at the contact patch, lessening the increase in tyre force that occurs due to weight transfer.
In between pedal strokes, there is less weight transfer and tyre force reduces. At the same time, the suspension returns to its normal position (the centre of gravity rises), which temporarily increases the tyre force at the contact patch.
An uneven pedalling action produces a relatively constant tyre force, resulting in more traction. 

Having less Anti-Squat is useful for ‘ironing out’ spikes in power delivery.

A bike squatting under acceleration can be considered as being analogous to a car having ‘body-roll’ when cornering.
Body-roll of a car when cornering is a useful mode of suspension movement. It allows cornering forces (initiated by steering) to be momentarily deferred, and imparted into the terrain over a greater period of time. In general, some amount of body roll provides improved traction when cornering. Too much body-roll, and the steering feels vague, and the vehicle geometry is compromised. Too little body-roll, and the vehicle will suffer loss of traction when cornering.

The same concept can be applied to the squatting action of a bike when accelerating.
Too much squat, and the acceleration will feel vague, inefficient, and the geometry is compromised (the bottom bracket moves closer to the ground, increasing the likelihood of crank strikes on the ground).
Too little squat (or some rise), and the bike could suffer loss of traction when accelerating. 

The above situations are by no means black and white. Most bikes will encounter both of these situations. This emphasises that there is no ‘perfect’ amount of Anti-Squat. It all depends on riding style, terrain, and rider preferences.

 

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Anti-Squat Curve (aka Acceleration Response):

The % Anti-Squat of a suspension system will often vary as the suspension moves. It is therefore useful to plot a graph of % Anti-Squat (y-axis) against Suspension Travel (x-axis), to produce a visual representation of the bike’s overall pedalling characteristics. This graph is called an Anti-Squat Curve, or an ‘Acceleration Response’, and is the ideal tool for comparing the pedalling behaviour of different suspension systems.

Earlier we mentioned the term ‘stability’, and defined a ‘stable system’ as one which has the ability to stay near its equilibrium position while transmitting power through the rear wheel.

Now we can go into more detail about exactly what a stable system is, by considering what happens when the system is moved away from its equilibrium position. In the case of a bike, the equilibrium position is sitting at sag. As the bike accelerates, there are all sorts of external interactions that cause the bike to move away from its equilibrium position, such as rider inputs and terrain inputs. Depending on how the bike reacts to these inputs, the system could be described as stable, neutral, or unstable.

  • In a stable system, a disturbance causes a change in the system, so that it automatically seeks to (fully or partially) restore equilibrium.
  • In a neutral system, the system does not seek to restore equilibrium.
  • In an unstable system, the system automatically changes in a way that causes it to move further away from equilibrium.

Most bikes on the market have Anti-Squat that decreases throughout travel. When accelerating with these systems, if a bump is encountered (or the rider inputs a ‘pump’), then the suspension compresses to a position with less Anti-Squat, meaning there is less force assisting the suspension to return to equilibrium. This type of system is unstable.

If the Anti-Squat increases throughout the pedalling zone, then when accelerating, if a bump is encountered (or the rider inputs a ‘pump’), then the suspension compresses to a position which has more Anti-Squat, meaning there is more force assisting the suspension to return to equilibrium. This type of system is stable. 

When riding a bike with a stable Anti-Squat curve, the bike feels a lot firmer under foot when pedalling, allowing the rider to run a lot less damping.

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Wheel Rate

Wheel Rate can be defined as the rate of increase in spring force at the suspended wheel, as the wheel moves through its travel.
Wheel Rate usually varies throughout suspension travel. Therefore it is useful to plot the quantity of Wheel Rate (y-axis) against the Suspension Travel (x-axis), to produce a wheel rate curve.
A wheel rate curve is a graphical representation of the spring rate felt by the rider when coasting.
Changing the shape of the wheel rate curve can significantly affect the feel of the bike.
Some words usually used to describe the wheel rate curve are ‘Progressive’, ‘Linear’, and ‘Digressive’.

The following describes how to calculate wheel rate:

Let the variable "x" be vertical wheel travel. e.g. for a bike with 200mm vertical wheel travel, "x" will vary from 0 to 200.
As the wheel moves through it's travel, the shock compresses an amount "s".
I.e. "s" varies as a function of vertical wheel travel "x". We can write "s" as "s(x)".
For all bikes, s(0) = 0.
And for a bike with 200mm vertical wheel travel, and a 9.5x3" shock (which has a 76mm stroke), then s(200) = 76.
For wheel travel values in between 0 and 200, the "s" values are dependent on the arrangement of the mechanism actuating the shock.

For any suspension system, these values of "s" can be measured throughout the range of travel (best done using software, such as Linkage, rather than physically measuring off the bike).
The values can then be plotted as a "Shock Compression" curve, with "s" on the y-axis and "x" on the x-axis. Linkage software provides this Shock Compression curve on the tab called "Geometry".
For most bikes, this will look like a relatively straight line (maybe with a slight curve) that passes through the origin. There is not much information that is visually obvious on this graph.

A much more useful curve, is the "Motion Ratio" curve. Motion Ratio is the derivative of Shock Compression (i.e. the slope of the Shock Compression curve). The Motion Ratio values are useful, because they tell you how much force there is at the rear wheel, for a known spring force:
Fw = MR * Fs

Therefore, if the spring force is known, and the Motion Ratio is known, then it's possible to plot a curve of Wheel Force as a function of suspension travel.
The Wheel Force curve is very useful, as it tells you whether the suspension is progressive, linear, or digressive. (+ve slope = Progressive, horizontal = Linear, -ve slope = Digressive).

NOTE: The bike industry generally uses the term "Leverage Ratio" instead of "Motion Ratio". In fact, Leverage Ratio is the inverse of Motion Ratio. At i-track, we prefer to use Motion Ratio, as it is a bit more intuitive. A larger Motion Ratio value means a larger Wheel Force felt by the rider. Whereas a larger Leverage Ratio value means a smaller Wheel Force felt by the rider.

Some people believe that the mere presence of Anti-Squat ‘stiffens’ their suspension under pedalling, making it less able to absorb impacts from obstacles in the terrain. This is not true.

However, if the % Anti-Squat varies throughout the suspension travel (acceleration response not a flat line), then there will be a slight change in wheel rate, when pedalling.
If the % Anti-Squat increases as a function of suspension travel, then powered acceleration (pedalling) will increase the wheel rate of the suspension system. The greater the pedalling force, the greater the increase of the wheel rate.
If the % Anti-Squat decreases as a function of suspension travel, then powered acceleration (pedalling) will decrease the wheel rate of the suspension system. The greater the pedalling force, the greater the decrease of the wheel rate.

This effect can be quantified, and is demonstrated in the following discussion:

We know that Anti-Squat generates an extension force in the rear suspension when pedalling (accelerating).
More specifically, we know that 100% Anti-Squat generates exactly enough extension force to counteract the compression that would occur due to weight transfer (definition of 100%AS).
The force due to weight transfer (in Newtons) is:



m = suspended mass (kg)
h = CoG height of suspended mass (mm)
a = forward acceleration (ms-2)
WB = wheelbase (mm)

Therefore if Anti-Squat = 100%, then the force at the rear axle increases by mha/WB Newtons.

We also know that if Anti-Squat = 0%, then there is no extension force in the rear suspension when pedalling (definition of 0% Anti-Squat)
Therefore, the force at the rear axle does not increase.

Based on this knowledge, we can construct a formula for the Wheel Force due to Anti-Squat (in Newtons):

Now, we can differentiate this with respect to x, to get Wheel Rate due to Anti-Squat (in N/mm):

Basically, this tells us that the wheel rate increases in proportion to how steep the Anti-Squat curve is, and how hard we are accelerating.

Now we can apply some real world values to quantify this effect.
Conveniently, for a bicycle, wheebase is roughly equal to the height of the CoG of the suspended mass (around 1100-1200mm), so 'h' and 'WB: cancel each other out.
Let m=80kg,
Maximum possible accceleration is approximately 0.5g. This is the point at which the front wheel starts to lift off the ground, and is a reasonable value to use for 'hard acceleration'. 0.5g = 4.9ms-2
So WRAS(x) = %AS'(x) * 3.92 (units: N/mm)

So, what's a typical value for the slope of the Anti-Squat curve? Well, it varies a lot from bike to bike.
Most suspension systems have an Anti-Squat curve that decreases as the suspension moves (negative slope). This means that under acceleration, the total wheel rate is decreased.
A few suspension systems (VPP, DW-Link, i-Track) are capable of having an Anti-Squat curve with a positive slope in a pedalling zone of the suspension travel. This means that under acceleration, wheel rate is increased.
(These suspension systems are also capable of having an Anti-Squat curve with a negative slope in 'non-pedalling' zones, to reduce pedal feedback).

During acceleration, weight transfer means that more of the suspended mass is carried by the rear axle.
One strategy might use this phenomenon to proportionally increase the wheel rate, such that the natural frequency of the system remains the same.

On our P1 Prototype, the slope of the Anti-Squat curve peaks at around 4.5 %/mm at around 20% travel, and is around 3%/mm at sag
So, back to our formula, under 0.5g acceleration, the Wheel Rate would increase by 3*3.92 = 11.8 N/mm.
This is significant, given that the Wheel Rate due to the spring/shock is around 8.4 = N/mm at that point in travel.
i.e. due to the shape of the Anti-Squat curve, the Wheel Rate more than doubles under 'hard acceleration'.

The same concept can be applied to the braking performance of a suspension system.
However, the forces involved are smaller, and therefore the effect is smaller.

The same formula can be applied to quantify the effect of Anti-Rise on Wheel Rate, when braking:

Maximum deceleration (due to rear brake only) is limited be the traction of the rear tyre, which depends on the coefficient of friction between the tyre and the ground. This can vary widely, especially when we are talking about mountain bike tyres and dirt/rocks/roots as terrain.
Assuming a 'ball park' value of 0.5 for the coefficient of friction, we calculate the maximum deceleration due to rear braking only to be 0.166g = 1.635ms-2 (if you're interested in learning how we calculate this, feel free to contact us via email).

We can use the same real world values as previously in the above equation:
Note: a is entered as a negative value, because we are talking about decelration (negative acceleration).
WRAR(x) = %AR'(x) * -1.308

Most bikes have an Anti-Rise curve that decreases as the suspension moves (negative slope).
An Anti-Rise curve with a negative slope will increase the Wheel Rate when braking.
Conversely, an Anti-Rise curve with positive slope (rare) will decrease the Wheel Rate when braking.
One strategy might use this phenomenon to proportionally decrease the Wheel Rate when braking, so that the natural frequency of the system is maintained.

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Braking Performance: Anti-Rise

We have seen how chain tension can be used to counteract the ‘squatting’ action that occurs due to weight transfer.
When braking, weight transfer occurs in the opposite direction. Weight is transferred from the rear wheel to the front wheel. Normally this would cause the rear end to ‘rise’.
Most suspension systems are designed such that braking forces cause a compression force in the suspension. This compression of the suspension can counteract the rise that would otherwise occur. This effect is called ‘Anti-Rise’.

Just like Anti-Squat, Anti-Rise can be quantified using a graphical method. Also, just like Anti-Squat, the quantity of Anti-Rise can be graphed as a function of suspension travel, to produce a ‘braking response’ curve.
The braking response curve provides a visual representation of how the suspension system performs under braking.

Anti-Rise is calculated as follows:

  1. Locate instant centre of ‘brake caliper member’ relative to front triangle
  2. Draw a line from the rear wheel contact patch, through point 1, and extend it over the front axle.
  3. Draw a vertical line through the front axle.
  4. Note the intersection point of line 2 and line 3, and compare it to the height of the centre of gravity, and the height of the rear wheel contact patch.

Many suspension systems have the brake caliper mounted to the wheel carrier member. With these systems, the braking response is dependent on the location of the pivot between the wheel carrier member and the chassis (real or virtual pivot point). As we have discussed, this pivot location also determines the wheel path and the acceleration response.
It is therefore obvious that a compromise is often made between key factors that we might like to design independently, because they are all dependent on the same feature (pivot location).

Some more advanced suspension designs (e.g. floating caliper, split-pivot, Trek ABP) have the brake caliper mounted on a structure that is not responsible for determining the wheel path.
These systems effectively ‘de-couple’ the dependency between the braking response and wheel path; allowing the braking response to be tuned independently.

This concept is analogous to the way that i-track suspension de-couples the dependency between the acceleration response and wheel path.

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