# Finally answered: String tension VS Room Temperautre.

Discussion in 'Badminton String' started by bigredlemon, Mar 1, 2003.

1. ### bigredlemon Regular Member

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Have you noticed how well your racquet performs changes from day to day, especially during swings of temperatures? As you know, the tension of the strings increases in colder surroundings because things shrink when it's cold. I haven't been able to find an answer on exactly how much this change is, so I’m going to working out the physics problem here on BF. My physics is a bit rusty so point on any mistakes if you see one. I'm pretty sure I've got everything right though. If you aren't interested in the calculations, just jump to the bottom for the answer. (My feelings won't be hurt... not at all... sniff.)

Theory:
When temperature changes, the length of the string changes.
Since the string cannot shrink itself as it is tied at both ends, the net effect it an increase in tension as though the racquet was strung with shorter strings and pulled extra hard to make it fit on the racquet.

Since the effective tension at the point of impact is different from when the racquet is "resting," the effective tension of a racquet in a colder room will be dependent upon the effects of the shift of the "resting" tension. A look from the stress-strain curve of Nylon 6 shows that an increase in stress will cause a greater proportionate percentage change in strain. This means the greater your default tension is, the greater the percentage effect this will have. Hence, racquets strung at high tensions are much much much more vulnerable to this temperature change phenomenon. Since I've never studied engineering, I'll just pull a fast one for this calculation and tell you that this phenomenon doesn't exist .

The numbers:
I wasn't able to find the actual specifications of badminton string, presumably because it is an industry secret, so I’ll be working with the closest polymer that's in the public domain: Nylon 6 Unfilled. I'll be using numbers published by Dow Chemical.

The typical coefficient of linear expansion for nylon 6 unfilled is 8.3x10^-5 mm/mm/'C (An mm of string will expand that many mm per degree change in 'C, or percentage change per temperature.)

Young's Modulus for nylon filament is 5x10^9 N/(m^2)

Since the elasticity is depend upon the cross-sectional area of the string, I will presume the string is 0.70mm wide. Thus, the radius is 0.70/2=0.35mm=0.00035m. The cross sectional area is thus 3.848x10^-7 m^2.

-------- WARNING: BORING ------------------------------------------------
Calculations:

E = FL0/AdL [young's modulus formula]
E = (F/A) / dL% [formula expressed as a percentage change in length]
dL% = (E / (F/A))^-1 [express formula wrt %change in length)
dL% = F/(E*A) [simplify formula]

c = dL%/T [identity of coefficient of linear expansion]
dL% = cT [re-express formula wrt %change in length]

c*T = F/(E*A) [combine two formulas to remove length altogether]

F = E*A*c*T [re-express formula as a function of string tension]

F=E*A*T*c [rearrange formula to make a cool word.. which is no easy... ]

F = 5x10^9 N/(m^2) * 3.848x10^-7 m^2 * 8.3x10^-5 (C^-1) * T ('C)
F = T(1924 N * 8.3x10^-5)
F = 0.159692*T (N)
F = 0.16*T (N)

Convert units into ones we use everyday:
0.16 (N) / 9.8 (m/s)/ 1 (kg*m/s) = 0.0163 kg = 0.036 pounds
1 'c *1.8= F

THUS.
One Fahrenheit change in temperature causes a 0.020 pound increase in string tension. Here's a table for your enjoyment:
1 'F = 0.02 lb
1 'C = 0.036 lb
1 'c = 0.0163 kg
1 'F = 0.0091 kg

--------- END CALCULATIONS -------------------------------------------------

What does this mean? The gym I play it gets really cold sometimes (cold enough for me to see my own breath,) which is about a 20'C difference between the temperature at which the racquet at and the temperature of the strings. This means that the tension of my racquet during playtime on cold days is actually 0.72 pounds tighter were I to be using monofilament Nylon strings. That's a small difference you say? Well, we don't use monofilament strings! Since I can't get my hands on actual data, I'll have to do some guesswork.

From the formula, we see that the tension difference is proportional to:
a) Young's modulus
b) Coefficient of linear expansion
c) Cross-sectional area of the string

C) the tension increasing effect is proportional to the square of the string size. A string twice as thin will increase the tension one-forth times as much. A string four times as thick will increase the tension 16 times as much.

B) Linear expansion is hard to predict. It heavily depends upon molecular structure, and even then, the pattern is not clear. It's why we have material enginners. I'd say, however, in general, materials more porous and soft will have higher values. Polystyrene has a linear expansion coefficient 60% higher. Acrylic which is hard, has a value 30% lower.

Zyex (used in Ashaway strings) for example has a value TEN times higher than nylon. a string made purely of Zyex would increase in tension by 7.2 pounds in my situation, and probably destroy my racquet. Vectran (used by Yonex) actually has a negative coefficient of thermal expansion, which means it actually EXPANDS when it's cold, and shrinks when it's hot. Since most strings are made out of more than one material, it's very hard to predict the total coefficient. Since most badminton strings feel softer than nylon. I’d guesstimate that the actual value is around 40% higher.

Btw, I suspect Yonex is using vectran partly to limit the effect of temperature on tension, thereby increasing the life of the string and maintaining consistency. Since most materials contract in the cold however, I suspect there is still some shrinkage occurs in Yonex strings overall. Nonetheless this effect would be much smaller, so great work Yonex !

PPS, Vectran is acutally made of carbon-graphite. Neat!

A) Young's Modulus is the ratio of stress over strain. It's kind of like how stiff something is. A string that stretches half as easily would have double the Young's modulus, and hence the temperature effect would be doubled. Since badminton string is usually not as stiff as a nylon string, so one could say this value is actually lower. On the other hand, badminton strings are multifilament strings, which makes the string feel softer without requiring it to be less elastic. This means the softness/bounciness is a matter of string constriction and not as a result of the material properties. Since most strings brag of increased strength, what Young's Modulus actually measures, I'd say the value I used is probably too low. The closest estimate is to use how durable a badminton string is compared to nylon. Since most string manufacturers brag about using materials that strengthen the string, I suspect the real young's modulus of badminton string is 200% higher. This means the temperature effect would be 200% greater than my calculations if we only consider this property. The difference between Young's modulus of nylon and polyester monofilament strings is 400%.

Btw, note that the tension increase felt by the string is determined not just by how much it stretches but also by how much "strength" it can exert per length it stretches. A rubber band can stretch a lot without changing the tension by a large amount, while a steel wire can exert an insane amount of force if it stretched equally as far (about a million greater increase in tension.) Thus, a company that brags that it's strings don't stretch as much during temperature changes is only telling half the story. If it's much stronger, than it could end up hurting your racquet more so than standard strings. Don't believe in marketing eh.

Someone mentioned awhile back that a racquet left in their car trunk was destroyed. I wonder what string it was strung with. Presuming the racquet was strung at 20 pounds in a 25'C/77'F room and left in the car at -20'C/-4'F using typical badminton string, that racquet would have been facing over 45 pounds of tension. No wonder it broke! Btw, since a thermal itself doesn't generate heat, it can only protect your racquet for a brief period before the racquet will be under pressure.

So in conclusion:
• Thin Vectran-reinforced badminton strings without a titanium core have a very small tension change in temperature changes, probably around 0.3 pounds in 5'C change.
• Thick Zyex-reinforced strings with a titanium core have a large tension change per temperature change, probably around 5 pounds in a 5'C change.
• Since a string manufacturer aren't usually both dumb enough AND unlucky enough to include all the "bad" features, your racquet probably experiences a tension change of around 0.5 pound per 5'C for Yonex strings that use vectran, and 1 pound per 5'C for all other badminton strings.
• Don't leave your racquet outside in the cold if you live a country like Canada, unless you can convince your stringer to do all the stringing outside in the snow

Notes: a misplaced bracket gave me a weird result (several million pounds of tension per degree change.) Finding that missing bracket actually took more time than writing this beastly report.

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Last edited: Mar 1, 2003
2. ### boyboy Regular Member

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OMG!!!!!!!!!!! :

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3. ### bigredlemon Regular Member

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Are you refering to that spelling mistake? I just corrected it now.

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4. ### nSmash Regular Member

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Good effort, BRL!

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5. ### eggroll Regular Member

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Nice job. Now, if we can get you working for J. Chretien!...

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6. ### ronk Regular Member

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Regarding the broken racket incident, I believe that the racket was not strung at 20lbs but somewhat higher. If it was strung at 25lbs, then the racket may have been facing tensions much higher than 45lbs. With such tensions, the racket will break so it might be the tension on the strings as I hypothesized rather than the temperature affecting the carbon graphite directly.

Great article -- now manufacturers need to provide some of the coefficents for all their strings so that players can calculate the tension on the string at different temperatures directly.

Ron

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7. ### Yodums Regular Member

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Sweet article dude.

Physics is really beneficial to life isn't it

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8. ### Neil Nicholls Regular Member

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I agree with your algebra and number crunching but my gut feeling is that 20lb doesn't turn into 45lb.
However, I don't live in a country that has such extremes of temperature, so I don't get exposure to the temperature effects that you do, so my gut could be way off.

I think a racquet would take 45lb tension if it is not being used and the tension gradually increases. Dynamic tensions on the racquet when it is use or while being strung should be more deadly that static loads. I don't know the details of Ronk's racquet so this is blowing in the wind a bit. Was it strung with 1 or 2 pieces of string.
If it was 2, then 1 piece of string may have snapped under the increased tension, and then the other piece of string could have collapsed the racquet.

If you have any string lying around you could do an experiment to find it's coefficient of thermal expansion. Take it outside when it's warm, stretch it out and mark on the ground how long it is. Do it again when it is cold and measure the difference.

On the dynamic forces side, if you hold a racquet head like a steering wheel, and then push your thumbs into the centre of the string bed (as if testing the tension)
you might be stretching the string by between 0.1 and 0.2 mm. This is 100 times the effect of temperature (I think). Could you crunch the numbers on that effect?

best regards,
Neil (physicist a long time ago)

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9. ### ronk Regular Member

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I don't know the details either, as it was not my racket. I vaguely remember reading in the thread that the tension might be high. I would not keep my rackets in my car trunk for weeks. I only leave my racket in my car for at most 1 hour like during lunch. I carry them racket bag up to my office and down again to avoid keeping the racket bag with several rackets in the car during extreme heat or cold.

Ron

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10. ### JChen99 Regular Member

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English plz!?

joking aside... Nice post and nice explainations... too bad I dun understand all of it... but kinda got a just of how it works now... thx BRL!

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11. ### bigredlemon Regular Member

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You're welcome. I dont understand all of it either.

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12. ### wilfredlgf Regular Member

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I really must commend you on your calculations, BRL. It could be made into an exam question for A-Levels/STPM by itself! Give yourself a good pat.

Too bad I can't really appreciate it - haven't touched Physics for over five years now.

Brought me back to the time when I did Form 6, this.

Can't comment on the matter though, I've been playing in hot halls (afternoon) for the past few months and I don't really see a difference when compared to playing under 'normal room' temperature.

Perhaps this is most useful to people who live in temperate regions of the world where the climate changes every three months, not really in tropical countries along the equator where the temperature difference is too small.

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13. ### iluvthesun Regular Member

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OMG A-level Physics ain't this hard!! I've only just finished GCSE Physics and I find I can cope with the stuff pretty well...

Anyway, your comment on play in hot vs normal conditions is understandable.

However, IMHO, the relationship between temperature and tension is not directly proportional (i.e. not a straight line). Between 20 and 35 degrees, there is less rate of change that between say -20 and 10 degrees. I notice that after leaving my Ti-7 (21 lbs) in temperatures below 0, not only does my hand get frozen to the grip , but I feel the tension increases quite dramatically as well.

I think that the non-linear relationship is due to the nature of the string construction.

BRL or anybody is very welcome to check this.

Mace

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14. ### ants Regular Member

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Man your good! Thanks for the effort. , I've learned something from this discussion.

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15. ### bluejeff Regular Member

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This is just too much for me

(no, I don't like physics, and that's why)

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16. ### bigredlemon Regular Member

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tension is calculated to be proportional to (absolute) temperature because of this:

solids are held together by molecular forces. The stronger the force, the smaller the object (per given weight.) Molecules vibrate and move, which counteracts this pull. This is why a certain amount of something can be larger than one another. The degree of this movement and vibration iis defined as temperature. Thus, temperature itself is defined as the "amount of stretching", all else equal.

I'm not sure why you wouldn't feel that is the case.

The temperature differnce between 20 and 35 is 15 degrees. The difference between -20 and 10 is 30 degrees. So you are right to say that a 15 degree difference shows less of a difference than at a 30 degree difference. It fits well with what i just said above.

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17. ### Sliced Drop Regular Member

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What about the effects of frame expansion/compression?

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18. ### bigredlemon Regular Member

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it's very small compared to the change exhibited by the strings and hence not considered. It should reduce the temperature effect proportionally...

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19. ### Sandman Regular Member

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Great effort! I admit to not understanding most of the physics involved, and don't care half as much since I live in a warm country. However, since most of the number crunchers are here, i would like to ask if you have any idea on how much tension is actually present during the IMPACT of the shuttlecock on the racket's stringbed? If pushing on the stringbed with your finggers increases the pressure on the strings, then the actuall impact of a full smash would also increase the string tension(albeit for a split second) considerably. Do you have any idea on how HIGH the tension actually spikes during IMPACT?

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20. ### bigredlemon Regular Member

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yes:
first determine the "strength" of the string: how much does the string stretch upon impact?

take that amount and put it into the equation i used in the first post.

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