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      04-14-2019, 01:04 PM   #1
fe1rx
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E82 Suspension Geometry

I have been working to create an accurate model of the E82 suspension geometry to allow me to understand its fundamental characteristics (bump steer, camber gain, roll centre, scrub radius, trail) as a function of those characteristics that can be tuned for performance purposes (ride height, camber, caster, toe). I have used a combination of CAD drawing and suspension analysis software (SusProg3D) to document and validate the results against real world geometric and kinematic measurements.

The process begins with establishing an appropriate coordinate system. Chassis measurement were made from an arbitrary point on the front subframe, with the chassis leveled longitudinally and laterally such that the tops of all four wheel wells were at the same level. This provides a repeatable configuration for measuring suspension mounting points in 3-dimensional space.

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To translate the chassis datum to ground level, I chose the ground plane as the vertical reference plane, the vehicle centerline as the lateral datum, and the nominal front of the bumper as the longitudinal datum. For modeling purposes, I have assumed a wheel centre ride height (wheel centre to fender lip) of 328 mm front and rear, and a tire rolling radius of 312 mm. Once the suspension model is created, changes in ride height and adding rake can easily be investigated, but starting with a constant ride height front and rear simplifies creating the model.

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Measuring of the chassis required removing all suspension arms and then the use of conventional and laser plumb bobs to transcribe the longitudinal (x-axis) and lateral (y-axis) positions to the floor, where they could be measured. The resulting measurement accuracy is expected to be ±1 mm.

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For bolted connections, target bolts were manufactured to provide the correct location of the associated ball joint or bushing centre.

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Similarly a fixture was built to locate the centre of the top of the strut tower.

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Measuring the rear suspension at the same time required fabricating a special fixture to locate the guide rod attachment point.

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The centre of the inboard rear camber arm bushing was found by first finding both ends of the bushing using a pointed bolt target and a laser plumb bob, and then finding the midpoint between those points.

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Once transcribed to the floor, the locations in x-y space could be measured.

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A laser level was used to measure the height coordinate of each point from a horizontal reference plane that could be referenced to the chassis datum height.

This is the general process I used to collect the required dimensional data. I will follow up with separate posts outlining looking at the front and rear suspensions in more detail.
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      04-14-2019, 01:13 PM   #2
fe1rx
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Front Suspension Geometry

The principal dimensions for the chassis related to the front suspension are shown in the figure below. For design purposes, front camber of -3.0° was selected, and a caster angle of 7.3°. These settings establish the strut top position for design purposes, which is assumed to be adjustable by means of a camber/caster plate.

Name:  Fig 9 Front Chassis Dimensions.jpg
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To complete the picture up front we also need the geometry of the steering knuckle, strut and suspension arms.

The true length of the suspension arms (ball-to-ball) requires some careful setup to measure.

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Name:  Fig 11 Front Arms.jpg
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Measuring the steering knuckle must be done with particular precision due to the close proximity of the various pivot points. This required disassembly of the steering knuckle, installation of spherical targets (recovered from actual suspension components to ensure proper geometry).

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Name:  Fig 13 Spherical Target.jpg
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A spherical target makes it easy to find the exact centre of rotation of each pivot location. I used a dial indicator to centre on each ball, and a digital readout on the vertical mill to find the location of each ball in x-x-z space. Desired accuracy is ±0.2 mm.

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The front strut has its own coordinate system (as specified in SusProg3D), centered on the wheel hub mounting surface. The assembly is measured with no camber or caster.

Name:  Fig 15 Front Strut Geometry.jpg
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With this data we can create both a CAD model and a SusProg3D model to visualize and characterize the suspension. The dual ball joint design of our front suspensions actually steers about a virtual lower ball joint that moves continuously with respect to the knuckle during both steering and vertical motion. Using CAD to find the virtual ball joint location for the one design condition is difficult but not impossible. Using it to solve for the location at a variety of suspension height is definitely impractical. Luckily SusProg3D handles this task with ease. Having both the CAD and SusProg3D models is useful to provide a cross check for logical consistency.

Name:  Fig 16 Front View Swing Arm.jpg
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To complete the model, the following additional data is required:

Initial Wheel and Tire data: I chose 18” x 8.5” ET45 wheels with tires with a rolling radius of 312 mm. These can easily be changed later in the SusProg3D model if desired.

Initial ride height: I chose 640 mm from ground to wheel well datum (corresponding to 328 mm wheel centre plus 312 mm rolling radius). Again once the model is established, ride height can be altered at will.

Suspension travel: I chose a bump travel of 40 mm and a droop travel of 66 mm, and chassis roll angles of ±2° as ranges of interest.

SusProg3D produces pages of data, but some of the main observations can be summarized as follows:

1) Scrub radius is between -6 mm and -14 mm over the complete range of ride heights and roll angles and is about -10 mm at ride height.
2) Trail varies from 19 to 29 mm over the complete range of ride heights and roll angles and is about 25 mm at ride height.

Name:  Fig 17 Front Roll Ctr Heights.jpg
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Name:  Fig 18 Front Cambers.jpg
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By adding steering rack geometry, bump steer curves can also be generated. I have not measured the rack with sufficient accuracy yet to provide valid bump steer curves, so that is on my to do list at some point. I am interested in looking at how changes in caster will affect bump steer.

The rear suspension is substantially more complicated than the front, and a few interesting nuggets have emerged from that analysis. I will follow up with those details when I get a chance.
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      04-15-2019, 10:55 AM   #3
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Massively impressed, and valuable data obtained. Thanks for doing this, it's going to be super helpful.
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      04-15-2019, 11:19 AM   #4
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It's going to be super helpful after I work hard enough and (maybe) can digest it all.

Many thanks to you for all the great work you done and posted!!
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      04-15-2019, 08:58 PM   #5
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Bravo, amazing work as always
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      04-24-2019, 09:34 PM   #6
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Assembling Data for the Rear Suspension Model

As with the front, we have measured the rear chassis pickup point locations with ±1 mm precision.

Name:  Fig 19 Rear Chassis Dimensions.jpg
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Similarly, the rear upright dimensions are required with ±0.1 mm precision. This required removing the upright (wheel carrier) and fabricating appropriate spherical targets to measure to. The upper, guide and toe arm targets were fabricated from the cores of discarded ball joints.

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Name:  Fig 21 Ball Joint Targets.jpg
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Initially I found the centre of the camber cross-axis ball joint and trailing bushing using a pair of 1” ball bearings as targets.

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Note that the OE location of the camber bearing in the upright is not centered in the lug. I call it “outset” from the center of the lug – i.e. away from the center of the upright. The OE location of the trailing arm bushing is similarly outset. The ability to adjust the locations of the bearing and bushing in their respective lugs is a potential tuning parameter that I will explore later. To further that exploration, I needed to know not only the center location but also the orientation of the axis so that I can find the bearing center at any arbitrary inset/outset. This prompted me remove the bearing and bushing from the upright and to use a 2.25” diameter ball bearing as a target on each side of the lugs allowing me to find both the center of the lug and its axis.

With appropriate targets installed, I found the upright Y dimensions of each target center using a height gauge and subtracting the target radius.

Name:  Fig 24 Upright Y-Dimensions.jpg
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I found the X and Z dimensions using a dial indicator to center on each target and a vertical mill with a digital readout to locate the coordinates.

Name:  Fig 25 Upright X- and Z-Dimensions.jpg
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The kingpin points on a multi-link upright being virtual, there is no natural “up” direction (Z-axis) to the upright. I recorded the brake caliper mounting holes and used these to define a provisional Z-axis direction. I later used SusProg3D to solve for the virtual king pin points at design ride height and alignment. The final upright geometry was then rotated to make the king pin axis vertical in side view (i.e. zero caster) to properly define the upright dimensions at zero camber, caster and toe. To allow for further investigation on the effect of insetting or outsetting the camber bearing and trailing bushing I have included these dimensions.

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We also need to know the pin-to-pin lengths of each of the rear suspension arms.

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Name:  Fig 28 Trailing True Length.JPG
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The camber arm has a nominal length of 400 mm, but this can be adjusted by means of an eccentric bolt to adjust camber. Accordingly, I treat it as a variable length arm.

The OE toe arm has a true length of 412 mm and its inboard mounting point is adjustable laterally by means of an eccentric bolt. Because I am using an adjustable length toe arm and have fixed the inboard mounting point, I am treating this arm as variable length also.

With these dimensions in hand we now have enough information to create a model in SusProg3D. I will follow up with a discussion of what that model reveals.
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      04-25-2019, 08:13 AM   #7
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All I can say is wow!
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      04-25-2019, 08:15 AM   #8
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      05-16-2019, 12:37 PM   #9
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Our rear suspensions consist of virtual upper and lower A-arms plus a toe link. Each virtual A-arm consists of a pair of struts, for a total of 5 strut elements. Although a rear suspension does not steer, it still has a kingpin axis about which it would steer if unrestrained by the toe link. Accordingly it also has a kingpin inclination angle (KPI), a caster angle, a scrub radius, mechanical trail and of course camber and toe angles. The following graphics are annotated versions of what SusProg3D presents after solving the geometry.

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The rear suspension has negative caster, negative scrub radius and negative mechanical trail. The why behind these design choices is interesting.

In contrast our front suspensions have positive caster (7.3°), positive mechanical trail (24 mm) and negative scrub radius (-11 mm), all of which are typical for front suspensions. Positive caster and trail result in a self-aligning torque about the kingpin axis in response to cornering forces. This is why steering effort increases with increased cornering g and why the steering wheel will tend to center itself if released in a corner. Negative scrub radius results in a toe-in torque about the kingpin axis in response to braking forces. This is a stabilizing influence under braking when combined with the unavoidable compliances in the front suspension. All these characteristics are desirable in a front suspension.

The self-aligning torque present in the front suspension tends to cause the front outside wheel to toe out in response to cornering forces. This torque will act with the compliances in the suspension to actually cause a small toe-out, and is therefore a stabilizing or understeer influence.

In contrast any toe out tendency in a rear suspension is a destabilizing or oversteer influence. Therefore self-aligning torque at the rear, acting with any compliance in the rear suspension is very undesirable. This explains why our cars have negative caster and negative trail in the rear. Both of these factors will result in a toe-in torque about the kingpin axis, which combined with any compliance will result in a stabilizing toe in rear steer reaction to cornering g. Due to a phenomenon called pneumatic trail, the side force generated by a tire does not act at the centre of the contact patch, but acts slightly aft of that point, and therefore is a toe-out or destabilizing influence. By taking all these factors into account, BMW has ensured that the rear compliance steer response will be toe-in (stable, understeer) and that the magnitude of the aligning torque at the rear is low and thus easily handled by the original toe link and its rubber bushings.

Negative scrub radius is stabilizing (toe-in influence) under braking at the rear, but destabilizing (toe-out influence) under acceleration so a bit is desirable for braking, but not so much as to be problematic under acceleration.

Negative caster is typical in rear suspensions and is readily apparent when looking at cars with strut rear suspensions. Internet chatter on this subject almost universally misses the mark as to why. Quite simply, it is to reduce loads on the toe arm and to ensure that any compliance rear steer is toe in and stabilizing in response to cornering g.

Incidentally, a multi-link suspension cannot be solved graphically. The virtual upper and lower king pin points are located at intersection of the upper and lower pairs of arms respectively when viewed along the king pin axis. But the king pin axis is defined by the virtual upper and lower king pin points. See the chicken and egg problem? You can only see the solution once you know the solution. Luckily SusProg3D solves the problem easily.

The main observations from the SusProg3D data for the rear suspension can be summarized as follows:

1) Scrub radius is approximately -44 mm (but can be adjusted to close to zero by insetting the lower ball joints in the upright).
2) Trail is approximately -40 mm at ride height.
3) Caster is approximately -8.3° at ride height.
4) Bump steer varies from approximately 0.1° toe in at full bump to 0.3° toe out at full droop. Slight toe in under bump is a stabilizing influence.
5) Roll center height as a function of roll angle is shown below. Unlike the front suspension, roll angle has little effect on roll center height.

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6) Roll centre height moves proportional to ride height.

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7) Rear camber gain with roll is greater than the roll angle, ensuring that the outside rear tire maintains a negative camber angle relative to the road.

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8) Rear camber gain accelerates with bump, unlike the less favourable response of a strut suspension.

Name:  35 Rear Camber vs Ride.jpg
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Last edited by fe1rx; 05-16-2019 at 12:42 PM..
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      05-17-2019, 03:08 PM   #10
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in your other thread you are talking about moving to spherical bushings in the rear and you reference M3 arms... is the above data base don the stock arms or the OE M3 arms? And you are lowered a bit, right?
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      05-17-2019, 04:56 PM   #11
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All of this data is so beautiful.
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      05-17-2019, 05:12 PM   #12
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Quote:
Originally Posted by bbnks2 View Post
in your other thread you are talking about moving to spherical bushings in the rear and you reference M3 arms... is the above data base don the stock arms or the OE M3 arms? And you are lowered a bit, right?
The SusProg3D model I have presented has an arbitrary ride height of 328 mm front and rear, which is not actually what I run. Once the basic dimensions are input and the model verified, data can be extracted at any ride height and alignment setting. So what I have presented is really just a typical example. OE and M3 arms are all the same length (as are the proplerly adjusted SPL arms) so the result apply equally to all arm variations. That said, the OE front wishbone is shorter than the M3 front wishbone, but that is the only exceptions and won't change the overall conclusions.
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      07-08-2019, 07:27 PM   #13
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Great work as always! Really awesome to see this modeled and characterised
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      07-11-2019, 09:55 PM   #14
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Really enjoyed reading this post!
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