A machine that holds its own load is always under half efficient
A standard screw thread converts 24.0% of the work you put into it and loses the rest inside the thread. Every bolt in the world is built to be that bad, on purpose.
Load the screw thread tool's M10 coarse preset: a 1.5 mm pitch cut on a shaft of 5 mm radius, holding 1,000 N, with a coefficient of friction of 0.15. The geometry offers a mechanical advantage of 20.94. The thread hands back 5.02, and the efficiency row reads 24.0%.
Three quarters of every turn goes into heat. Show a colleague a gearbox that did that and it would go back to the supplier. Nobody has ever sent back a bolt.
Friction, measured as an angle
μ arrives in school as a ratio of two forces, which makes it something you need two instruments to see. Take its arctangent and it turns into a slope you can measure with a protractor.
Put a block on a plank and lift one end until the block starts to go. That angle is φ = arctan μ, the friction angle, and it holds everything μ does. At μ = 0.15 it is 8.531°. Below that slope things stay; above it they slide; and the weight of the block never comes into it, because gravity pulls the block down the slope and presses it into the plank in the same proportion.
This is the form of μ worth carrying around, because it can be compared directly with the slope of a thread.
The thread, unrolled
Take one turn of the screw and flatten it out. It climbs by the pitch, 3 mm, while travelling once round the shaft, 62.83 mm. The tool draws that triangle beside the cylinder and reports its slope: a helix angle of 2.734°.
2.734° against a friction angle of 8.531°. The thread is nowhere near steep enough to let go, which is why the bolt is still there in the morning. The tool prints the comparison as tangents on the row labelled self-locking: 0.0477 against 0.15.
The price and the guarantee are one line of algebra
Switch to ramp mode, where the arithmetic is cleaner, and put the two readouts side by side. Efficiency there is the actual advantage over the ideal one, which reduces to
η = tan θ / (tan θ + μ)
and the holding condition is tan θ ≤ μ. Now substitute one into the other. If tan θ is at most μ, the denominator is at least twice the numerator, so η is at most one half. If tan θ exceeds μ, η exceeds one half. At the boundary the efficiency is exactly 50%, and the tool will show you both sides of it: set μ to 0.30 and a ramp at 16° holds, at 48.9%, while one degree steeper slips, at 50.5%.
I swept 106,253 ramp states through the tool's own module, every angle from 1° to 89° against every μ up to 1.5. The holding verdict and the efficiency never disagree once.
A machine that will hold its own load has already given away more than half of the work you put in. That is one inequality, read from either end.
The screw obeys the same bound with a slightly different constant, because the effort on a screw is applied round the shaft rather than along the thread. Sweeping 16,520,845 self-locking screw states, the most efficient of them reaches 49.9905%, and it gets there only on a nearly frictionless thread whose helix sits exactly on the locking boundary, one notch of pitch from letting go. Real threads are nowhere near that line. The Standard Screw preset sits at 24.0%.
What it rules out
Of the six machines in the simple-machines chapter here, the inclined plane and the screw thread are the two with explicit holding conditions. Their tools put that verdict beside the efficiency. A lever, a block and tackle and a spur gear train all run backwards the instant you let go, which is why every one of them that has to hold something is fitted with a pawl, a brake or a pin. The screw keeps its brake in its geometry and pays for it in the only currency available.
So there is no design meeting in which somebody gets both. A screw jack that holds a car in the air cannot also be pleasant to wind, and a thread pleasant to wind will not hold the car. Choosing a self-locking machine is choosing to throw away most of the input, and the machines chosen that way are exactly the ones nobody wants running backwards: the jack, the machine vice, the worm drive on a hoist, the bolt.
The same inequality, in an axe
Switch the tool to wedge mode. The condition becomes tan(α/2) ≤ μ, with α the included angle, because a wedge is two ramps back to back and only one of them faces each half of the log.
At μ = 0.25 that boundary is an included angle of 28.07°. The tool's 15° preset sticks fast, at 16.7% efficiency. Open the wedge to 30° and it no longer holds, and the two halves push it back out.
Which is the difference between a felling axe and a splitting maul, and it is not a difference of weight. A thin axe buries itself in a green log and has to be worked free. A maul is ground blunt enough that the wood rejects it, which is precisely the behaviour a bolt is designed never to have.
Where the account runs out
Everything above is static: one load, one direction, nothing moving. Bolts come undone anyway.
Gerhard Junker's 1969 paper is about the case that does it, and the loading is across the bolt rather than along it. Once the clamped faces begin sliding sideways against each other, the thread is momentarily relieved of the friction the static condition assumes, and the bolt walks out a fraction of a turn per cycle while tan λ ≤ μ remains perfectly true. His rig became the standard transverse-vibration test, written into DIN 65151 and now ISO 16130.
This is where being tight finally earns the credit it is usually given for the wrong reason. Preload is not what holds the thread; the helix angle does that, and it does it at whatever tension the bolt is under. Preload clamps the joint hard enough that the faces never slip, which is what keeps the static condition applicable in the first place. A bolt tight enough to self-lock and not tight enough to stop the joint moving is a bolt you will find on the floor.
References (2)
- preloaded joints coming undone under vibration, which the static holding condition cannot predict Junker, G. H. (1969). New Criteria for Self-Loosening of Fasteners Under Vibration. SAE Technical Paper 690055.
- the friction experiments behind μ being a property of the surfaces rather than of the load Coulomb, C.-A. (1785). Théorie des machines simples, en ayant égard au frottement de leurs parties et à la roideur des cordages. Mémoires de Mathématique et de Physique, Académie Royale des Sciences.