Friday, January 6, 2017


Photo: Thomas Mielke
One of the great things about Twitter is its ability to act as the proverbial office water cooler for a lot of us who no longer have steady access to an office water cooler; it's a mix of weird gossip, pompous pronouncements, hissing provocations, and all the other good stuff that comes with being around people, with the added bonus of being able to select just who you're likely to bump into at any one time (depending on retweets which originate from various uninvited and unsavory third parties, sincere intention to spark discussion notwithstanding).

And like any proverbial water cooler, in that mix of gossip and pomposity and so on occasionally something genuinely interesting pops up.

So a bit more than a week ago we got this:
Yeah, it sounds like the kind of question that would come up amongst a bunch of gearheads passing a bong and listening to Ummagumma-era Floyd in some off-campus basement. (Maybe that's actually what Twitter is?) But it's also a really good question.

In fact this is a great question, a gloriously iconoclastic inquiry in a world currently obsessed with Can-Am-grade outputs. Forget about the flood of recent mills that have made cars with 0-to-60 times in the four-second bracket seem lackluster. Set aside the Hellcats and the Lamborghini SVs and the ludicrous-mode Teslas and the rest. What's the least motor you need to get by in the world?

Of course the answer is the same as the answer to the perennial question about what the Greatest Car In The World is: it depends*. Or, as quickly came up in the conversation:
Right. Of course this kind of rulemaking has a habit of getting into a deep ontological discussion about what a car actually is and how ridiculously tiny and flimsy a structure one can propose and still take seriously, never mind the inevitable mind-expanding dive into the possibilities of gear reduction enabling a Cox .049 motor to 'technically' motivate a GMC 3500HD or whatever. But those both get to the absurdum part of a reductio ad absurdum way too quickly to be useful and eventually you're better off just admitting that the ultimate solution is to lose the motor completely and ride a bicycle, which is not a bad thing at all but also not the answer.

If we're going to take this seriously - and this is a question well worth taking very seriously - let's consider those parameters that Wojdyla brings up and fill in some sensible judgement calls:
  • First, let's make this a real car and not a Peel P50 or go-kart or powered barstool or something like that. Yes, we'll obviously have to skew small here, but we also want this to be recognizable as a car and (mostly) work in the real world. So small two-seater or very small four-seater, which matters more as weight (and to a lesser extent air drag) than sheer size.
  • Second, let's make this at least passably functional in the real world, which means that our hypothetical States of Motion MicroMotor Special will need to be able to achieve, oh, an arbitrary but not irrational 20 meters per second (that's 72 kph/~44.74 mph), which at least is good enough for suburban streets and country lanes if not Interstates.
Yes, we're going to mostly be using metric through this; it's much easier to perform calculations. I'll throw in a few converted figures when necessary. And there will be a bit of discreet rounding here and there, but it should all work out in the end.
  • Third, to simplify the whole process a bit we will disregard concerns about packaging and displacement and instead arrive at a solution based solely on output, which tends to generally dictate the rest of that anyway.
  • Fourth, nothing else that we normally consider when thinking about a car - handling, design, safety, comfort, cruising range, long-term reliability, appeal to potential copulative partners - matters in the least here. We are strictly concerned with motivating a box capable of carrying a few nearly-normal-sized humans from one place to another.
So let's think about what it actually means to power a vehicle based on some elementary physics.

In idealized steady-state driving, such as cruising along a flat and level highway, the vehicle only needs enough power to overcome various forms of drag, mostly air resistance and the deformation and stickiness of tires on the road surface, and some subsequent driveline friction. Find a way to reduce or eliminate that air resistance and tire drag - think magnetically-levitating hovercraft moving in a vacuum - and our vehicle acts as a pure inertia device which requires zero energy to keep moving. This doesn't work in the real world**, of course, but we can design this thing to be pretty slippery - and at 72 kph it's not going to be facing that much of a wall of air anyway - and we can also go ahead and specify ultra-low-rolling-resistance tires, maybe stolen from a first-gen Honda Insight or something. So barring a shipping container being dragged behind the car, cruising is not a major issue.

I'll even say that acceleration isn't a huge issue here either. Yes, we need this thing to convert a certain amount of potential energy of some form into kinetic energy as it gathers up its skirts and eventually works its way up to our 72 kph, but we won't have a mandatory minimum acceleration parameter. Any continuing application of force sufficient to overcome drag will result in acceleration (basically force = mass times acceleration - or, as we'll use later, acceleration = force/mass - with some minor parasitic factors) and we'll have an adequate amount of that given how this is something which will get resolved in the course of dealing with our biggest concern.

That biggest concern is what happens when that road isn't flat and level. We do not live on a giant ball bearing, and hills are a fact of life***. The SoM MicroMotor Special - oh, what the hell, let's call it the Mouse - will have to pull itself up an incline to be considered a functioning automobile, and this is where the fun starts.
Photo: Andreas Praefcke
Let's add another bullet point to our list of Wojdyla parameters:
  • The vehicle must be able to climb an arbitrary 7% grade of indefinite length while maintaining a speed of 16 m/sec (57.6 kph/~35.8 mph).
And for this to occur the motor must do honest-to-Archimedes work and deliver it at a set rate, which means we need to have a certain definite minimum power figure, which can be calculated.

Notebooks out.

So some terms need to be clarified: First, we're concerned with weight instead of mass. Mass is a measure of how much matter something contains; weight is that matter under the influence of gravity. A kilogram really doesn't weigh one kilogram, but rather 9.8 newtons***.

Work is the act of moving an object by applying force. More commonly, and more relevant to our concerns, it's the act of lifting a certain weight a certain distance or rotating an arm (or a wheel, which is just kind of a continuous arm if you think about it...hey, anything left in the bong?) around an axis, which we all know as that glorious thing that is torque.

Power is the act of performing that work in a certain time.

Let's get our parameters set up in real units:

A 7% grade is almost exactly 4 degrees from horizontal. That doesn't sound like much until you have to climb it; a 7% grade is the maximum permitted on Interstate highways and even then only for short distances in mountainous terrain (a 6% maximum is more generally enforced). For visualization purposes, 7%/4° converts to a rise of 1 meter for every 14.3 meters in straightline distance.

We next need to consider the weight of the vehicle. Naturally the Mouse will be as light as is practical, but again we're aiming for at least some sort of real-world relevance so yes, a real structure, and no, not a carbon-fiber skeleton wearing a Reynolds Wrap skin. Quick research shows that most tiny people movers with (ostensibly) four seats - the Tata Nano, the original Issigonis Mini - scale in at about 600kg. Weave in some Japanese engineering genius and a small parallel-twin motorcycle engine and you have the Honda N360, which weighs about 500kg and which we'll accept as a reasonable minimum for something that won't fold in on itself*****.

Photo: 天然ガス
That's just for the vehicle; at the very least we also have to include a driver. Yes, we could hire some winsome and fawnlike ultralight Eastern European fashion model to be our test driver, but let's again aim for a certain real-world element. Propriety says that to cover various arrangements of passengers and luggage and drinks in cupholders and whatever we should skew the driver's mass slightly large, towards the Matt Farah/Jack Baruth/yr hmbl svt end of the body mass index, and so we'll add 100kg****** as our operator/payload mass.

That gets us up to 600kg that has to be hauled up this 7%/4° grade at the mandated 16m/sec by something more than good intentions.

The equation to find the required power turns out to be very straightforward, especially when using those metric units:

Weight (in newtons) times rate (in meters per second) times distance (vertical component only since that's what the work here is doing so sine of 4°) equals power in watts.

(Yeah, I also thought it was too simple at first as well, except that a watt is defined as (kg × m²)/sec³ so that cleans it up in a hurry.)

Plug in our Wojdyla parameters of 600kg vehicle mass and 16m/sec speed up the incline, and using the accepted 9.8m/sec² constant for gravity and rounding sin(4°) from 0.069756474... to a more palatable .07:

600 × 9.8 × 16 × .07 = 6585.6 watts

Or 6.6 kilowatts, or 8.85 horsepower. Add 10% or so for mechanical losses and various invocations of the Second Law of Thermodynamics and we can say that the Mouse really needs no more than 7.5 kilowatts, or 10 horsepower.

That's not much as far as motors go, even in the pre-Hellcat era.

Just for the sake of it let's figure out the resultant torque and acceleration.

Two things: First, given a constant power output the motor can either spin faster with less torque or spin slower with more torque. A motor producing x torque and turning at n rpm will make the same power as one producing 2x torque and turning at n/2 rpm. This conversion will happen repeatedly below.

Second, one important note about calculating this and what it means to the vehicle itself: In order to know how much force is being applied to the pavement, we naturally need to know how much force the motor is generating at any one instant. This is not an easy thing to do with internal-combustion motors; because of cam timing and ignition curves and a bunch of other factors they do not produce consistent torque throgh the rev range, which anyone who has ever looked at a dyno chart knows. If we fitted the Mouse with a CVT we could theoretically have it run at a consistent ideal speed, but it's better to go ahead and use a power source which produces wonderfully steady (if subjectively boring) torque, run through a constant ratio.

Yes, it's an electric Mouse.

Okay, so those wheels and tires we stole off of someone's 2000 Insight are sized 165/65R14, so they have a circumference of just about 179cm. In order to make the Mouse move at that mandated flat-surface speed of 20m/sec they have to rotate 11.17 times per second, or 670.39 times per minute.

10 horsepower being delivered at 671 rpm [HOLD ON IMPORTANT UPDATE HERE: We're NOT spinning the wheels at 671 rpm to get 10hp; that power calculation was done at 16m/sec so we actually have to figure this at 537 rpm - corrected numbers follow] translates to a decent 78.3 agreeable 97.8 foot-pounds of torque, or 132.6 joules, delivered consistently at the contact patch. (The motor will spin a bit faster; redlines on  commonly available 10hp electric motors tend to be around 1700-1800 rpm. If we run it through effective 2.6:1 reduction gearing, that means the motor is making about 37.6 foot-pounds, or 51 joules. Seems about right.)

And by the way, this 25% correction means that the Mouse's motor now should make about 12bhp at max revs, given how torque tends to fall off a bit at top with electrics.

The radius of the wheel-tire combo is 28.45cm (1/3.515 of a meter) and 132.6 joules is 132.6 newton-meters, so the drive wheels are pushing the Mouse forward with a more or less constant force of 132.6 × 3.515 = 466.1 newtons.

Acceleration is force divided by mass, so 466.1n/600kg for the win.

The Mouse will accelerate at a generally consistent rate of 0.78 meters per second, per second. It will take a little something over 26 seconds to hit our mandatory top speed of 20m/sec - but it will eventually get there.

Yeah. Um...that's slow*******. (Not as slow as with the miscalculation based on 671 rpm, but still.) Admittedly if we used a multiple-ratio gearbox we could use that magical reduction gearing from very early on to get more force to the ground and clip a substantial amount of time off of this figure. Something to maybe work on later.


Having gotten through all this, let's look at some parallel real-world/historical examples of barely-adequately-powered vehicles, even if none of them are electrics:

Photo: Daimler-Benz
We might as well start at the beginning. The Benz Patent-Motorwagen produced a whole roaring 2/3 of a horsepower (500 watts) from its near-liter of displacement but did so at all of 250 rpm, which means that torque at that engine speed was a slightly more palatable 14 lb/ft or 19 joules. (And it probably needed every bit of it to rotate those big carriage wheels from a start.) Even so, it was enough to get Berta Benz and her two teenage boys through a near-200km round trip to see her mama at about 15kph, which was audaciously fast in 1886.

The N360 from earlier was overpowered for our purposes; its 354cc parallel-twin spun out 23kW, or about 31bhp, and top speed was an excessive 105kph/~65mph.

Photo: Thomas Forsman
The closest equivalent to the Mouse in real life is probably the Citroën 2CV, which weighs about 600 kg and in its early 375cc form cranked out 9 hp/6.7kW - although as with the Benz that would likely translate into a somewhat less existentially troubling torque figure. Unfortunately, its top speed of 65 kph was apparently adequate when meandering from charming Gallic village to charming Gallic village in the 1950s but doesn't measure up to our modern requirement. (Later ones were faster, although the idea of "fast" when talking about 2CVs is always a bit relative.)  The electric motor might provide a bit more oomph, though, and its super-short 1st gear apparently gave it some impressive grade/stair-climbing potential.

So that's what you need. Would anyone actually want a Mouse, though? No, not really. The inability to operate at Interstate speeds is a massive handicap, general around-town effectiveness would be marginal at best, and again we haven't thought about anything else that goes into making this thing the slightest bit likeable.

Absolute minimum real-world power is probably along the lines of a 36hp Volkswagen Beetle, which again does better if you measure torque; in more realistic terms the 68hp of a three-cylinder Mitsubishi Mirage is about as low as anyone is willing to go to propel a modern car. Even the very Mouse-like Mitsu i-MiEV makes 63hp from its electric motor, and no one thinks of that as a rational all-around vehicle.

But I do sometimes think it would be cool to own a 2CV, though.


*: Of course this answer is a total evasion. Everyone really knows that the Greatest Car In The World is an Alfa Romeo Giulia TZ2, unless you unfortunately have to carry more than one other person with you and in that case it's a Mercedes W123 wagon. But no one wants to say this all the time because most normal people tend to get that kind of furrowed-brow thing going on when you say this, so it's better to punt.

**: Although I'm certain someone in Silicon Valley thinks otherwise and will be hitting up venture capital shops to fund this particular exercise in disrupting an anachronistic status quo as soon as they figure out how to put it in an app.

***: No, not that kind.

****: Residents of Kansas may beg to differ.

*****: Seriously. Even early Lotus Sevens weighed around 500kg (and those occasionally folded in on themselves).

******: By happy coincidence also the normal weight of an average adult male Ailuropoda melanoleuca, or giant panda.

*******: Maybe we should find one of those Eastern European models.