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I would also like to find more information on the Mumford link as it seems to be a potentially useful modification to my Bugeye Sprite. Can you point me to any sources of information on how it works? There doesn't seem to be a great deal of info readily available, perhaps it's not as popular as a Watts linkage of Panhard rod because of the complexity and difficulty in determining link sizes, pivot locations, etc. Any leads you can give me will be greatly appreciated.
The Mumford link, or linkage (since it is a linkage: a system for motion control containing more than one link, together with other elements), is actually a family of possible layouts, commonly used for lateral location of beam axles. This family has at least four possible members.
There are two other families. The better known Watt linkage consists of a central rocker, and two links attaching to the ends of a centrally pivoted rocker, extending in opposite directions from the rocker. The WOB linkage has a rocker with its pivot at one end, and two links different distances from the pivot, extending in the same direction from the rocker.
The Mumford linkage uses two rockers and three links. All three systems can give very nearly straight-line axle location in ride, while allowing roll about a reasonably well defined point, which serves as the roll center, or as one end of an axle axis of rotation whose axle plane intercept defines the roll center.
The Mumford linkage is thus a more complex means of accomplishing a simple task than its rivals are, so the only way it can be justified is if it either provides better packaging or load paths in a particular situation, or if it permits geometrical properties unobtainable otherwise.
The usual appeal of the Mumford linkage is that it is able to provide a roll center below any part of the hardware. The roll center can be lower than the ground clearance of the car. It can be lower than a smooth floor pan under the rear axle, transitioning smoothly into a diffuser.
Now, why do we want an unusually low roll center with a rear beam axle? Not to prevent jacking. Maybe to minimize lateral scrub on one-wheel bumps, particularly with stiff tire sidewalls. But more commonly, we want a low rear roll center with a live rear axle so we can have a lot of elastic rear roll stiffness, via springs and/or antiroll bars, relative to the front elastic roll resistance. The advantage of this, for a road racing car, is that it minimizes torque wedge Ė the change in diagonal percentage due to driveshaft torque.
With appropriate design, there are better ways to minimize torque wedge. But these may or may not be understood by the designer, and they may or may not be allowed by the rules.
It should be noted that there is also a way to get a roll center below any portion of the linkage using a Watt linkage. More on this later.
There isnít any standard nomenclature for the parts of a Mumford linkage, so I am inventing some. The mechanism has two rockers, each with two link attachment points and one pivot. One link from each rocker connects to the axle (if the rockers are attached to the frame or sprung structure) or the frame (if the rockers are on the axle). I will call these the side links. The third link connects the two rockers to each other, and constrains their movement so they can only pivot equal amounts, in opposite directions. I will call this the center link.
There are four basic configurations of the Mumford linkage. The rockers can be near the center of the vehicle, with the side links extending outward from them, and a short center link, or they can be out toward the sides of the vehicle, with the side links extending inboard from them, and a long center link. Additionally, the rockers can be on the frame or on the axle. Thus, there are four permutations:
∑ Rockers unsprung (on axle), inboard (RUI)
∑ Rockers unsprung, outboard (RUO)
∑ Rockers sprung (on frame or sprung structure), inboard (RSI)
∑ Rockers sprung, outboard (RSO)
All of these options can give a roll center below any point on the linkage. However, they differ with respect to what I call the Mitchell index: how the roll center height varies as the suspension moves in ride.
When the rockers are unsprung, the roll center only moves a little with ride, relative to ground. It moves oppositely to the sprung mass: a Mitchell index that is negative, with a small absolute value.
When the rockers are sprung, the roll center moves with the sprung mass in ride, a bit more than the sprung mass: a Mitchell index a bit greater than one.
If the system is paired with an independent suspension at the front, the front suspension will in most cases have a Mitchell index of one or greater. It is difficult to get a Mitchell index near or less than zero with independent suspension.
It is desirable to have similar Mitchell indices at the front and rear of the car. This way, the roll axis inclination will not change drastically as the car negotiates humps and dips while cornering: the relationship of front geometric anti-roll to rear geometric anti-roll will not vary a great deal with heave displacement. The carís oversteer/understeer balance should therefore be more constant, particularly in undulating sweepers. That lets the driver press closer to the limit of adhesion, with less worry that the car will get away from him.
However, many cars race fairly effectively in total disregard of this. It is not the end of the world if the front and rear Mitchell indices are dissimilar.
I mentioned that it is possible to get a very low roll center with a Watt linkage. This involves laying the rocker flat, under the axle, and having it angled in top view, rather than straight front to back. The two links are then made higher at their outboard ends than at their inboard ends, where they connect to the rocker. Care has to be taken to make sure the joints donít run out of angular travel as the system moves. Packaging permitting, the rocker can be mounted to the sprung mass, and the outer ends of the links can be attached to the axle tubes. In that case, the Mitchell index is a bit greater than one: better for compatibility with independent front suspension, but not essential. Alternatively, the rocker can be attached to the axle, and the outer ends of the links can attach to the sprung mass. In that case, the Mitchell index is negative, and small in absolute value.
The questioner is interested in applying the system to a Mk. 1 or Bugeye Sprite. The stock suspension on that car consists of trailing quarter-elliptical leaf springs a bit below axle center, and rubber-bushed trailing links above those. The springs provide lateral location. There is no additional mechanism for lateral location. The roll center is then at spring height, and the system appears to be fairly compliant laterally.
American production sports car rules (e.g. SCCA) would require that the leaf springs be retained, but would allow addition of lateral locating devices, and also allow changing bushing design.
If the objective is merely to take lateral compliance out of the system, probably the best approach would be a diagonal Panhard bar running approximately from where one of the leaf springs connects to the unibody to where the opposite spring connects to the axle. This would work with stock bushings, or with upper links having rod ends and adjustable length. It would not lower the roll center. The bar might need to have a bend, and pass over or under the driveshaft.
If a lower roll center is desired, the attachment of the springs to the axle has to be modified so that the springs no longer locate the axle laterally. Or, rules permitting, the springs might be eliminated entirely, and replaced with Heim-jointed links. In any case, the axle must be free to move laterally with respect to the springs, or the system will bind in roll.
If the roll center is lowered, the car will need more rear spring or anti-roll bar for similar steady-state cornering behavior. The main advantage to going this way will be that the car will put power down a bit better exiting right turns.
TOE OUT FOR TURN IN
Popular wisdom always suggests that for a faster turn-in, toe-out is the way to go. I've never read a satisfactory explanation into the possible reasons and it seems counter-intuitive to me. Thinking about what happens at the tires, if you have toe-out on turn-in the outside tire has to pass "over center" before it begins building grip in the direction of the turn. Thinking that the outside tire is carrying most of the load and therefore creating the majority of the grip I would think that toe-in would provide a faster turn-in as you would be building grip faster as the more loaded tire would already have a slight slip angle before you even turn the wheel.
I've never done much on-track experimentation to see if I personally believe toe-out improves turn-in, but it's hard to believe such a prevalent opinion came out of nowhere. I have a theory that maybe the cause could be the change in relative wheel heights as the scrub radius/caster kingpin/trail cause the inside front wheel to move down in relation to the chassis and the outside to move up.
It would seem that with a fast enough turn in that the inside was temporarily the heavier loaded tire and its steered angle would create more grip than the outside until the point that the load transfers to the outside tire.
If this is the case, it would seem that in softly sprung vehicles with lots of steering inclination, toe-out would be the way to go, at least on tight courses requiring fast turn-in.
What are your thoughts on this age old myth/truth?
Cars do generally exhibit quicker initial turn-in with static toe-out. My analysis is that this does not have to do with the lateral (y axis, per SAE convention) forces from the front tires, but rather the longitudinal (x axis) forces, which can also produce yaw moments.
When the car is running straight, and the front tires have either toe-in or toe-out, the tires are both running at a slight slip angle, and accordingly generating both some lateral forces and some drag forces. The drag forces are roughly equal, and additive. The lateral forces are roughly equal, and opposite in direction, so they approximately cancel.
When the handwheel is turned just a tiny bit, one front wheel will be running straight, and the other will be turned into the corner, generating a bit of drag, and some lateral force into the corner.
If the car has toe-in, it will be the inside front wheel thatís running straight, and the outside one that has some slip angle. In this condition, the lateral force creates a yaw moment into the turn, but the drag force creates a yaw moment out of the turn: the two yaw moment components are subtractive.
If the car has toe-out, it will be the outside front wheel thatís running straight, and the inside one thatís making lateral force and drag force. Now the lateral force and the drag force both create yaw
moments into the corner, and are additive. Consequently, the net yaw moment is greater, and the car experiences a greater yaw acceleration: it turns in quicker.