On Standard Screw Threads

 On Standard Screw Threads

(Part 3 of Nuts & Bolts)

I am still pursuing the question "Do modern metric nuts and bolts come loose spontaneously with greater facility than the nuts and bolts of my childhood; and, if that is the case, why are the modern nuts worse than the British Standard Whitworth nuts of my youth?". 

When I raised the question 15 years ago (See ref. [1])  I baulked at the problem of calculating the effect of 'groove-angle' on (a) friction, and (b) the the force causing the nut to unwind. But I pointed my finger at the increased groove-angle as possible culprit for the present casecade of falling nuts. However, buoyed by my success 4 years ago in calcultaing the packing density of uniform spheres [2], I now venture to tackle the 3-dimensional mathematics of the bolt, and its helical thread [3]. 

I suppose a bolt could have a helical groove with a square section, and indeed there may be some special applications in which such threads are cut. However, the groove of the standard bolt of 19th and early 20th century British engineering had a symmetridal, v-shaped profile, where the angle beweet the upward facing surface and the downward facing surface was 55º. Now, in the 21st century, internatinally, that angle is 60º. 

90º⊐  55º (>  60º (> 

(Image from boltscience.com)

Once tightening is complete, there will a tension in the bolt along its axis related (by Hook's law) to the elastic stretching of the bolt. That tension (F, in the sketches below) will be countered by pressure on the surface of the (upper) bolthead and the surface of the nut in contact with the lower plate contact area. But F will also be felt by the entire surface area of the bolt thread that is in contact with the mating surface of the nut, where it will generate friction. It is that friction, proportional to the surface area in contact and to the force pressing them together, that stops the nut from unwinding. 

Let me consider first the square-sectioned groove (sketched in Fig. 2 below.)  In the middle-top sketch I have tried to draw a square-bottomed grove on the right side of a right-handed bolt, trying to show that, as the thread comes towards us, it dips by a small amount below the horizontal; I have suggested 5º (A pitch of 1.27mm in circumference of 19.95mm gives a 'lead-angle' of 3.65º). In the middle-bottom sketch I have tried to draw the more normal v-bottomed groove, with a similar small dip of 5º. 

(Fig. 2)

In the left-top I have suggested that the Hook's law tension (F) operates along the axis of the bolt. It is met, of course, by the nut. I have supposed that the friction that develops between bolt and nut is proportional to the area of the two mated surfaces, and to the force between them that acts normal to the mated surfaces. I have likewise supposed that the force that unwinds the nut operates in the plane of the nut. 

From the sketch in the left-top, it would seem plausible that, in the case of the square-bottomed groove, the normal to the bearing surface is only 5º off vertical; leaving most of the force (85/90) remaining to generate friction (and a residual 5/90 of the force to unwind the nut). In the case of the v-bottomed groove it would seem that a large portion of the of the force F operates at the wrong angle to cause friction, 27.5/90 in the case of the BSW nut, 30/90 in the case of the modern ISOmetric nut. (I have not managed to see what that tangental force does; does it stretch the nut? )

Thinking along the above lines I have managed to convince myself that the v-bottomed groove does not maximise the friction between nut and bolt, because the tension in the bolt is not normal to the mated surfaces. And in this regards the modern bolts of the ISOmetric series with a 60º groove are worse than the British Standard Whitworth (BSW) with 55º groove. Incidentally the American UTS series also uses the 60º groove. 

Perhaps a more significant difference between the BSW groove and the shallower ISOmetric groove is the smaller area of the mated surfaces in the latter. To show how this could be significant, I have sketched (Fig. 3) an equilateral triangle (sides 2, 2, 2; angles 60º, 60º, 60º), and a second isosceles triangle where one of the angles has been narrowed to 55º. You will see that the friction-generating surface of the latter is 8% larger.

(Fig. 3)

These two effects combined might explain my conviction that the nuts of today are more inclined to come loose than the nuts of the 'Whitworth' era. 

(I have also shown that a small element of the elastic tension in the bolt is directed towards un-twisting (loosening) the nut. It is dependent on the ratio of the pitch to the circumference. But in that regard the nuts of the ISOmetric series are very similar to those of the BS Whitworth series, and are occasionally 'finer' (See my Nuts and Bolts1). I have not yet managed to clarify whether the tension in the bolt applied to the sloping side of the groove could have a resultant component that un-twists the nut, like the progress of a sailing boat with a beam wind. I would urge engineers like those at  boltscience.com  to investigate these questions. )

REFERENCES

[1]  https://occidentis.blogspot.com/2010/06/nuts-and-bolts.html 

[2]  https://occidentis.blogspot.com/2021/09/excluded-volume-between-close-packed.html

[3]  https://occidentis.blogspot.com/2025/06/on-helices.html

Iran

 Iran

I went to Iran once. Thirty three years ago, so back in the early nineteen nineties. I was invited to give a series of lectures at the Tarbiat Modares University in Tehran, jointly with a postdoctoral scientist from Oxford. A fascinating country, and a charming and worthy people. 

Why do I bring this up now, you may ask. Though I fear I may have left it too late, I feel it is my duty to share my experience. I suspect most people in Britain will be as near to totally ignorant of Iran as I was before my trip; ignorant of both the geography and the history of the country, and of the people. 

        I was invited, I suspect, in an attempt to forge links between our universities; more that I might learn about Iran than that their students might learn about 'Transport across Biomembranes'.  I returned to the University of Newcastle conscious that I was meant to be an advocate for closer relations between my university and an equivalent university in Tehran. I tried, but found that our pro-vice chancellor for overseas students had got it into his head that Islamic countries tended to default on paying fees. Some past problem with Libya, I think; possibly related to the Lockerbie bombing. But how stupid to confuse Maghreb arabism with Aryan Iran; as though Islam taught commercial duplicity, which is pretty rich coming from 'Perfidious Albion'. I indignantly protested that Iran had a very good reputation for business probity, but to no avail. I was in any case pretty sick of my university, and life generally at that time; and I left it at that; a dozen lovely memories, and as many photos. 

The Iran Air plane, an enormous Jumbo-Jet, filled up with well-dressed Europeans, like any other international airliner. The only oddity being signs announcing the times for prayer, and the direction of Mecca. However, no sooner had we taken off than the women all produced and donned head-scarves from their handbags. How would I recognise my fellow lecturer, who I knew was on the plane somewhere? It turned out not to be a problem for, after touring around the giant plane for 5 minute, I saw coming towards me a figure who could only be the Dr. Saira Malik I was looking for, a bright young Oxford biophysicist of Pakistani heritage, albeit wearing a headscarf. She seemed as confident as I was, that she had found what she was looking for: a British biochemist/biophysicist in mid-career and a fellow Oxonian. 

After chatting awhile and making a plan to link up on arrival, we returned to our seats; in my case to read the highly relevant book I had bought at Heathrow airport – "What is Islam?" [1]. That brought me swiftly from bottom of the class to a midway position able, in rural UK, to pontificate on  the 114 surahs of the Qur'an, Fatima, Ali and the Shi'ah, the role of the Jewish bible, relationship with Christianity; the Assassins, the Whirling Dervishes, and Sufi mystics. I looked nervously about me as I read of the Satanic verses, and the murderous, 1300-year-old, vendetta between Sunni and Shia sects. I marvelled at the enthusiasm these people bring to their beliefs. 

Our Hotel was the old Hyatt Hotel, lying just off the Chamran Highway in the extreme northern part of Tehran city, commandeered and renamed the Azadi Grand Hotel in 1979, along with other American assets. I cannot remember how we were met at the airport. But the morning after our arrival we were taken by car a mile or two south to Tarbiat Modares University. Saira and I gave our lectures alternaltely; she on membrane fluidity, I on the bioenergetics of solute uptake.  We lectured on a raised stage to slides projected onto a screen behind us. After the lecture, cups of sweet tea were handed round to all, starting of course with the lecturer, who then took questions. 

We had free time. I made cautious scouting trips round the blocks near the hotel, enchanted by early April almond blossom in the quieter streets, little artisan bakeries making the flat-bread discs that are their starchy staple. And, gazing northward to the majestic, snow-covered, mountains where the Shah had built a ski resort, before he hurried off to the States in 1979. On one of our free afternoons, academic colleagues took us walking into the city; leafy streets, with a runnel of clear water next to the pavement, and a welcome paucity of cars. (The revolution of 1979 had interrupted the import of new cars. However, the traffic down in the lower city was admittedly rather polluting.) 

On another free afternoon we were taken up above the snow-line on the ski-lift to the Shah's chalet. There we slid down a snowy slope sitting on an inflated truck-tyre. 

(Image by WorldAtlas.com)

Most enjoyable for me was a walk by myself from the hotel out into the foothills of the Elburz Mountains. There were well made stoney tracks following up the river valleys. The terrain was extremely rocky, the business solely recreation. There was no cultivation, and no sign of grazing animals; scarcely a green leaf, except for some poplar trees down by the rushing water. There was a sprinkling of recreational walkers, including women clad in black from head to heel. Every now and then I saw a mule carrying a load presumably to a restaurant further up the valley. It was idyllically peaceful. The Iranians drank great quantities of what looked like 'Coco-Cola' but was presumably made locally to their own recipe. The restaurants offered neither tables nor chairs, but a low platform on which, after removing your shoes, you could sit and eat, enjoying the dappled shade of the trees. (I learned some years later from an Iranian student that, for an ordinary Iranian at home, it is perfectly normal to sit on the floor to eat.)  I found it wonderful to be able to walk straight out from my hotel deep into the wilderness, and back, in an afternoon. The almond blossom, the rugged path, the rushing water, the mules, the women clad in hijab – I had never felt so close to Omar Khayyam.

One of my tutees in the Biochemistry honours course at Newcastle was from Tehran. She had asked her parents to call for me at my hotel and show me round Tehran. Neither speaking English nor owning a car, her parents had nevertheless taken the trouble to hire a car with an English speaking driver. I saw some lovely views of the city, and for a few miles out on the road northwards, towards the mountains and the Caspian Sea. But I was also invited to their home, where I was offered some hospitality. I greatly appreciated their kindness, and this brief opportunity to meet another culture, at home.

Sixteen months later I was able to repay their kindness for the mother of my student came over to see her daughter receive her degree. I do hope mother and daughter enjoyed the  coast and castles of Northumberland as much as I had enjoyed my week in Tehran. 

REFERENCES

[1] "What is Islam?" By Horrie & Chippindale, 1990, Virgin Publishing, London

My Funny Turn

My Funny Turn

       I had a funny turn the other day; on Thursday evening, at half past six.

    I was bustling from a pre-opening Private Show at the Museum ("Photographer of the Year") the short half mile to the Writers' Café on Parson's Street. So, across the canal, through the car-park, down the snicket and round into Parson's Street. I was nearly there, when I noticed that the ground immediately in front of me had begun to sway around. Not just the ground. The corner of 'The Old Auctioneer' pub was also swirling around. A bollard-thing, guarding the corner, lurched towards me and  I grabbed onto it. Just as well, for I believe I would otherwise have tumbled over in the street. 

     There I clung for 10 or 20 seconds, as thing gradually calmed down. Then, letting go, I tentative walked the last few yards to the Café. Never, in my life, before or since (it is now 10 weeks since) have I felt anything like it. 

       Reason tells me that it was I, and in particular my head, that was swaying around, and not the world. But I want to emphasise the subjective experience, the ego-centric, eye-centred, feeling of stasis. It seemed as though, for the space of 10 or 20 seconds the information from my vestibular systems of the inner ear was switched off. My eyes told me that the road was approaching my face. My vestibular systems should have detected an acceleration of my head towards the road, but detected nothing, reporting,  instead, the condition of being stationery. 

      Was I fainting from cerebral anoxia, or hypoglycaemia? I noticed absolutely no 'greying' of the visual field that I associate with fainting. I normally eat at 6.30 but, having enjoyed a beer at 5.00, I had skipped dinner. I reckon I could live off fat reserves for days or weeks. But I realise that the automatic maintenance of blood sugar, might get faulty with age; everything else does. I am not sure that I have ever been hypoglycaemic and do not know what that feels like. So I asked an acquaintance with type 1 diabetes melitus, and found that I had none of the symptoms.

      When I got to the café, I had a coffee and a flapjack. I felt fine, completely normal; and now (ten weeks later) have never had a recurrence of that 'funny turn'.

Should I be concerned? Should I give up driving?

On Helices

On Helices

(Part 2 of Nuts & Bolts)

You will recall that some 15 years ago [1] I raised the question "Do modern metric nuts and bolts come loose spontaneously with greater facility than the nuts and bolts of my childhood". I was a little surprised to find no one taking up the enquiry, as it surely affect us all. (By the way, the handle is loose again on the cold tap in my new kitchen, thus defeating a handyman and two trained plumbers!). So, I decided to look further into the matter. 

One useful step forward was identifying a company (BoltScience.com) with a testing jig that will rattle a couple of bolted plates till they loosen. They have not yet (so far as I can see) asked my questions, about metrication, tribology, and groove angle; but I may be able to cajole them into a colaboration. As a second step, I decided to make a deeper study of the helix, because its geometry is fundamental to the questions I am raising. 

At the outset, let me clear up a possible confusion: a spiral staircase is a misnomer. A spiral would lie of the floor. We should talk rather of a helical staircase; c.f. Watson & Crick's double helix. Next, let us remind ourselves of some terms. 

Let h = the (vertical) height of the helix up its long axis, 

Let p = the pitch or lead up the long axis for each turn of the nut. 

Let L = the length of the helix were it unwound and laid flat.

Let c = the circumference of the helix.

Let n = the number of turns; i.e. h/p.

I next wanted to see if I could calculate the area of the two surfaces that are juxtaposed when a nut is threaded on a bolt, the two surfaces between which the friction is generated that prevents the nut loosening; and for that I must calculate L, the length of the thread. 

Not trusting my theoretical powers, I sought to measure L, to check theory. I found my wooden rolling pin (h=400mm), milled to a uniform diameter of 39.8mm which I determined by wrapping a paper strip round the cylinder and cutting it exactly to equal  the circumference (c=125mm). I found that a dressmaker's measuring tape would lie adequately flat against the rolling pin, even when wound round like a ribbon on a May-pole.   

Two extremes are easy. If the tape runs vertically up the pin, L simply equals h. And if the tape went round and round the pin without progressing up the pin at all, L could be infinitely long. I measured the length of tape needed to wind it round one full circle in the 400mm height of the rolling pin. And again with 2 full circles; etc.  

No. of complete circles.  Measured L (mm)

1 417

2 472

3 552

4 646

5 750

With string I could have explored further, but the tape was more convenient; and, lying flat against the rolling pin suggested the next step. With the tape making one complete cycle in the height of the pin, it was easy to see that the rather stiff tape made a precise angle (𝛉1) against the vertical axis, which it maintained all the way round the pin. I could find L by applying Pythagoras's theorem to the little right-angled triangle: base = 𝛿c; height = 𝛿h; hypotenuse = 𝛿L.; and assume that the relations held all the way up. Furthermore, the angle off the axis (𝛉) can be calculated by applying trigonometric functions.  For a single circle of the rolling pin, the cos of the angle (𝛉) of the thread to the axis is 'adjacent/hypotenuse' = 400/417, so 𝛉 = 17º; for two circles of the pin the angle between thread and axis (𝛉) is 32º [cos(32º)=400/472]. Likewise, the related small angle (90º - 𝛉) is arctan (pitch/circumference). 

For each circle of the pin:

L² = c² + p²

But n (the number of turns) = h/p so, for a multi-turn helix we can predict: 

L = h/p x √(c² + p²)

I add these predictions to my table. 

No. of  circles.  Measured L (mm) Calculated L (mm)

1 417 419

2 472 470

3 552 548

4 646 640

5 750 742

That is all very satisfying. And is a step towards calculating the area of the two mated surfaces between nut and bolt where friction prevents loosening. 

REFERENCE

[1] https://occidentis.blogspot.com/2010/06/nuts-and-bolts.html

Blackbirds or Pigeons

 Blackbirds or Pigeons

I turned to the woman on the seat behind me as the bus waited at the traffic lights. We had recognised each other. She normally remembers my name, which is flattering. On this occasion I also remembered her name.

"Are you coming home from working at the pharmacy?" I asked, though it was mid-afternoon on a Saturday. 

" No! From the gym." she told me, and now that I noticed it, there was a touch of pride as well as of tiredness in her voice. 

"A therapeutic workout?" I suggested, to which she grudgingly assented.

What is life for, after all? You work till you are tired, raise kids, grow old and die. She helps people already, by dispensing medicines, but would dearly like to become a qualified counsellor, and listen instead to people's woes. It is amusing how small a shift in perspective will change a chore into a calling.

"And I should like to travel", she told me. 

I mused a while. Should I tell her about the blackbirds and the pigeons? Would it amuse her, lighten her load?

A number of bird species live in or near my small garden, and carry on their business there.

Robins, wrens, dunnocks, tits and sparrows flit about, more or less unseen. I occasionally glimpse fluttering wings, or hear a snatch of song from deep in dense foliage. Starlings strut about on the ridge tiles opposite, chattering their comments on the scene below; but they seldom come down to the garden. So, it is mostly the larger birds that concern me here; the blackbirds and the pigeons. 

I have often watched a hen blackbird picking raspberries off the canes I grow in two giant pots against the sun-soaked fence. She crouches on the flag-stones looking up at the canes above her till she finds a ripe berry, then leaps up and snatches at the fruit to fly off with it in her beak. I am fairly relaxed about her depredations. When I go down and take a look, I find no fruit worth picking; nothing except tiny, unripe, fragments. She must be hungry. Perhaps she has a nest-full of chicks nearby. 

The male blackbird is often seen on the lawn, ear cocked as though to hear the rasping noise of subterranean worms, or swooping low across the lawn, with a long, thin, worm in his beak; further evidence of hungry nestlings. Blackbirds are territorial. Each pair has its garden, which they must defends against poachers. Hence the valiant singing with which the males fill the gloaming; that evening stillness between sunset and nightfall. Blackbirds strike me as good parents. 

Not so the pigeons. I have never seen a hungry pigeon. They always seem well-fed, and idle; showing only a dilettante's interest in a poppy seed on the flagstones or a couple of ants on the lawn. They are strong flyers and can feed anywhere in the parish. I resent their sleek plumpness, seeming to be perfectly aware that they are now a protected species. And I resent even more the amount of time they spend love-making. I don't mind when they sit quietly 'kissing'; it seems touchingly human. But I dislike their posturing and crooning and, above all, the noisy flapping of their wings as the cock repeatedly tries to tread his hen.  Or when he indulges in showy combat with rival cocks.

  "But the essence of life is finding food, and rearing a family" I suggested, "like the blackbirds. Not just strutting about making love, like the pigeons". 

"Humph", she retorted. 

"Swallows and swifts are great travellers", I mused. "There are almost none nesting here this year, but I saw great numbers of each last week, in Croatia."  

"I would really love to travel", she repeated.     

Behind the Hedge

 Behind the Hedge

I have an old violin I bought in 1972 and used frequently for playing in pubs and folk clubs. Its distinguishing feature is scratched into the varnish of the back – 2£. But it plays freely. The low clearance at the nut makes stopping easy. Through frequent use I am familiar with the curve of the bridge, and the separation of the strings. It has history**. But last summer it came unglued, and was inexpertly repaired by a wife-husband team who deal mostly with school violins. And now the pegs do not hold the tension of the strings. Caswells Strings is a much bigger concern with a well stocked showroom in town. My niece bought a violin from them 7 years ago and, more recently, I bought a 'mute' there. But I had not seen any sign of a 'craftsman' on their premises.  

I looked online for local violin makers (as I had some years ago in Mexico City – see my blog-post), and found an entry from a firm I had never heard of, in a tiny hamlet I had passed through many times; a mere handful of houses and a letterbox. I telephoned to make an appointment and was given instructions on how to find the "workshop", by turning off the lane 50 yards before the hamlet. 

I could not have been more surprised if I had found behind the hedge, in that little hamlet, a fully equipped and well manned space ship. As the door opened I heard a faint hum, as from the engines; two men in aprons; it could be about to take off.

One man was bent over the body of a double bass gripped in a large vice and was preparing to fit a new neck to the block. The other I must have caught between jobs, for he had nothing in his hands at the moment I entered. I looked around me, astonished.

It is amusing that the firm advertises itself as 'Violin Makers', for they clearly specialise in the less romantic double-bass, and more often repair those than construct them from scratch. The founder of the firm had for many years been a professional bass player with a London orchestra till he retired some 30 years ago, to open the workshop. The 'firm' moved up from Henley 25 years ago. 

        There were some twenty basses lining the walls in various states of repair, but also a cello in the process of construction and a solitary violin, ready varnished but as yet unplayed.  

'David' looked at my pegs, applied chalk to all, gently reamed one peg-hole, trimmed a whisker off its peg to make it more circular in section, and drilled a new hole for the string further up the shank. All inside 40 minutes.  

(** If you are interested in the story of how I came by my old 2£ fiddle, drop me a line to: cawstein@gmail.com)