On The Geometry of Sidecars by Peter Watson

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In my last little polemic on side car stuff, I looked at some 'hows' of building a basic set of Earles type leading link forks. This time we'll have a go at some whys and wherefores of sidecar design geometry. The first thing to get a grasp of is the asymmetric nature of the beast. There are forces pulling and pushing in all directions. The best way to approach it is with an end in mind. Surely the object is to have something that rolls in a straight line and can be directed with minimal and predictable effort. Anybody who has had a crack at it will have run up against just how much of a black art this simple quest can appear to be. The asymmetry at its simplest, gives power delivery on one side and drag on the other. The steering is also on the power delivery side. The tendency is therefore for the bike to drive around the outfit wheel in a circle. Not too flash. To counter this less than desirable tendency there are a number of things that need to happen. Figure 1 (After Brice 1977) is from a motorcycle service manual and shows the desirable amount of toe-in. Toe-in is the slight difference in track of the bike and the outfit respectively. Distance 'A' is approximately 3/8" to 3/4" ie 10-20mm greater than distance "B". The effect of this is to push the front of the bike, in this case, to the right. The trick is to do so by the same amount as the dead wheel wants to pull the whole machine around to the left. This sorts us for rolling in a straight line on flat ground. But roads are not flat ground ! Bugger! Roads are in fact cambered with a crown down the middle. This is for very sound engineering reasons and aids drainage and helps separate the opposing streams of traffic. To overcome camber we come to our next trick. At this point a quick aside may help some aspirants with their three wheeled addiction.

A bicycle is actually quite a complex little devise. To balance and steer it requires a whole set of circumstances which deserves an entire article (or two) on its own. To compress that whole article, the sequence of events that gets you around the corner is a relationship between gyroscopic procession, castor (or trail) and the rolling cone effect. Pointing the wheel is pretty much a secondary performance. In fact, to get the sequence happening at all, at anything over walking pace, you actually start by pushing the handlebars in the opposite direction to that which you desire

Figure 1: Toe-in. After Price, 1977

to turn! Don't believe me? Try crossing your arms, ie right hand to left bar etc and consciously try to steer. You'll end up on your backside if not your nose in the dirt, with a sense of bafflement like that which you had when first you tried riding, back in the Archean (well, for some of us anyway). What does all this have to do with camber and who let this prat loose on the word processor? I hear you say. Well, to overcome camber we need to induce the rolling cone effect! And that isn't something in a bottle!

Figure 2: Lean out. After Price, 1977

Figure 3 (from Foale and Willoughby 1984) is the easiest way to describe it. In short, what you do is to make the bike fall over. Gyroscopic procession actually helps induce the fall in the opposite direction to that which the wheel was at first turned. The castor makes the wheel follow the point at which the steering axis meets the ground. The induced lean produces a rolling cone as can be seen in Figure 3. The wheel thus moves through a circle, ie around a corner, as a result. A similar effect occurs when you drop a wheel into a rut or tram track. It wants to climb out, up the side of the slot. A road can be thought of as though the rut wall was horizontal and the wheel leaned right over. It "climbs" across the road in a curved path or around a corner. Combined with rider input, the bike self centres under the castor effect and gyroscopic procession and thus rights itself after cornering.

Figure 3 (from Foale and Willoughby 1984)

So now to the 'lean-out' bit. If you stop and think for a moment you will see that camber is just a very shallow, single-sided rut. What the lean-out does is induce the bike to "climb" across the road pulling the sidecar with it, a sort of complimentary cousin of the toe-in effect described earlier. To achieve lean out we need to have the plane of the bicycle wheels leaning away from the vertical, away from the side carrying the outfit. Figure 2 above shows the desired amount on a motorcycle with suspension etc. The amount needed for a bicycle is of a similar proportion, though I tend to use a little less due to limited suspension travel. "C" should be less than "D" by ~15-25mm. This should do the trick perfectly. Again, give yourself a bit of fudging room when assembling it all and before you hit the arc welder or gas for the last time. It can be seen that a bit of suck-it-and-see type juggling will in all likelihood reach a functional compromise. How do you design it in? In reality you probably can't. What you do have to do is to design and build the beast with a bit of adjustment left in it. This is what I was alluding to in my last article when I was talking about the placement of the outfit wheel mounting points in the chassis.

In Figure 1, you may have noticed, the outfit wheel leads the rear wheel of the bike. Why? So where does one put the contact patch of the outfit wheel? Good question. When you corner any vehicle you are inducing centrifugal force. The outfit wheel needs to be placed so that the centrifugal moment of the centre of mass of the vehicle (including you and any passengers) goes through (or as near to through as possible) a vertical line from the contact patch of said outfit wheel, and preferably the contact patch itself.

If the centrifugal moment of mass is in front of the wheel, there is an inclination to sweep the front wheel sideways and the rear one off the ground. Radical under-steer, increasing with speed and or decreasing radius of the corner, can be rather disturbing, not to mention digging the front edge of the outfit into the ground. Hmm. If the centrifugal moment is behind the outfit wheel, the front wheel is lifted off the ground and, as above, swept sideways. Again this is a rather disturbing result which worsens with speed and decreasing radius. Control becomes a serious problem. Indeed the very hoons pictured in the previous article have flipped theirs several times at speed. And that is on a machine that has had quite some effort put into getting it right and a pair of nutters who know how to move their weight about. Mum can't watch. The three wheeled asymmetry is by it very nature a compromise. But fun! What you need to do is work out the sorts of loads you intend the thing to carry and place them so that the combined mass of load (stuff and or passenger/s), rider and combo itself) all sits roughly within the plane of the outfit axle. It will always be a compromise, though you will know if you get it badly wrong. I use small (20") wheels for the outfit. There are 12-gauge spokes readily available for this sized wheel. The radius of a 20" wheel is minimal and thus subjects the axles, spokes, chassis etc to much lower leverage forces, especially when cornering with the outfit on the outside of the corner (turning right for the examples above). It will survive longer. Trail or caster is something I whizzed over last time, possibly presuming a bit much. There are a couple of issues in play here. Firstly, with a two track vehicle, there is not the imperative to have the front wheel follow the leaning bike so strictly. Conversely a short castor on a single track vehicle gives very quick handling at the expense of induced lateral oscillations. Anybody who has pushed a piano with a wheel that flicks from side to side will know an extreme example. The two track vehicle will not induce the oscillatory effect to the same degree, thus reducing it as I recommended to about 40-60mm still provides self centring without the need to physically drag the contact patch around the steering axis. The longer the castor the greater the leverage required and thus the greater the need for Armstrong Steering. A complicating factor is Rake. Rake is the steering head angle. If there is no rake, and the axle is directly below axis, the wheel would turn on the point at which the axis meets the ground. There would be no Castor. If the axle is behind the vertical steering axis the would be castor equal to that distance behind. The point being that rake and castor are not mutually dependant per se. If there is rake, as is usual, then as the wheel is turned, the distance between the axle and the ground at the contact patch, is shortened as the wheel is in part leaned over. This is known as steering head drop. A good analogy is that of a ball on a mound (See Figure 4). Virtually no force is required to keep the ball in place at the top; if it is allowed to drop however, we must lift its weight to restore balance. This is a condition known as unstable equilibrium (Foale & Willoughby 1984). In other words EFFORT, your effort, is required to keep, or more accurately return, the front wheel to the centre. Not really what you want in the circumstances. Along with a steepish rake, the shorter castor also reduces this effect markedly. Hence the reason for building the leading link forks in the first place. Lateral strength and the opportunity to use suspension are also bonuses worth having.

Figure 4: Unstable equilibrium. After Foale and Willoughby, 1984.

Figure 4: Unstable equilibrium. After Foale and Willoughby, 1984. A final point for now is the placement of the fork pivot and the brakes. If you go back to the original drawings and photographs in the Sidecar Design Lab article, you will see that the pivot axis is directly opposite the brakes on the swinging arm and both are at the rim radius. This has the effect of directing the turning moment induced by the brakes and the spinning wheel, directly through the pivots.

If the brakes are behind the pivot, the bike rises around the pivot. If the brakes are in front of the pivot, the opposite effect is induced and the front end is lowered under braking. Neither help with control, braking, or suspension function. In fact the suspension is required to deal with the induced movement. The example I have given you allows the front suspension to travel normally over the road surface, unhindered by the effects of braking. The wheel base remains stable, steering is unaltered and braking remains consistent even when cornering. For those who want the real thing vis a vis technical advice, you can do a lot worse than get hold of a copy of Tony Foale's very readable and accessible Motor Cycle Handling and Chassis Design: The Art and Science. Published by Tony Foale Designs. It is also available in Spanish. There are a number of other very good engineering focused publications around, again largely on motorcycle design, which you will find if you are keen. Cheers, pw References: Foale, T & Willoughby, V. 1984. Motorcycle Chassis Design: the theory and practice. Osprey Pub. UK. Foale, T, 2002. Motor Cycle Handling and Chassis Design: The Art and Science, 2nd Ed. Tony Foale Designs, Spain. http:/www.tonyfoale.com/ Price, B. 1977. BSA 500 & 650cc Unit Twins; Service, Repair, Performance. Clymer Publications, LA.

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