Monday, May 26, 2014

Energy density of a spring

Energy density of a spring (or a "mainspring")

My 2-year-old son loves wind-up toys, and that set me thinking... what is the energy density (the energy per unit mass) of a wound-up spring? After a bit of googling, I have come to the conclusion that this is one of the few questions to which the internet does not know the answer!

A clockwork toy's gearbox A clockwork toy's mainspring, fully wound down.

Quite a few people have already asked the question.

For example, on the xkcd forum " today I began thinking about mainsprings, the coiled springs typically used to power wind up clocks, watches, etc. While reading up on them I began to notice a trend where articles comment on how much energy they can contain (usually described as "a lot" rather than anything useful.) This led me to try to find a source for the potential energy of a mainspring, something that I've found rather difficult to find." The same question has been asked more than once on physicsforums.

And there must be plenty of experts who know the answer

... for example these Birmingham researchers, and about 400 years of clock-making experts, and Trevor Bayliss who made the wind-up radio. I don't know why they are so secretive! :-)

Let's figure out a rough answer to the question

A coiled spring stores energy in the same way as a bent beam. You can read about the energy stored in a bent beam in my lecture on the musical note produced by the "beams" of a marimba or xylophone.

Gamelan - a musical instrument like a marimba

It's interesting stuff, but actually we don't need all that detail to get the answer. The key insights we need are

  1. The energy per unit mass in a bit of the spring that is strained with a strain of ε is
    0.5 Y ( ε2 ) / ρ
    where Y is the Young's modulus, and ρ is the density.
  2. The stress τ is (roughly) related to the strain by
    τ = Y ε
  3. and the maximum stress you can cope with [in a spring that is to be reused many times] is called the Yield strength, which I'll denote by the symbol τmax.
Putting these facts together, if εmax is the maximum strain [εmaxmax/Y] The maximum energy per unit mass in the bit of the spring that is maximally strained is
0.5 Y ( τmax/Y )2 / ρ = 0.5 (τmax)2 / ( Y ρ )
Now, not all of the spring will manage to be in this state of maximum strain, and the spring must be housed in a container too, which adds to the weight, so this energy density is an upper bound on what is possible with a spring made of a given material.

Material Y τmax ρ 0.5 (τmax)2 / ( Y ρ )
GPa MPa kg/m3 (Wh/kg)
Steel (structural ASTM A36 steel) 200 250 8000 0.005
Titanium 116 225 0.013
Rubber 0.03? 16? 1100 1.1 ?
Carbon fibre [.] 230 4000 1600 6.0?
Carbon nanotubes ? ? ?
Steel (Micro-Melt 10 Tough Treated Tool (AISI A11)) 200 5000 7450 2.3

We can compare these energy densities with those of other energy storage systems featured in my book by looking at page 199. Sadly, the wind-up spring doesn't get close to the energy density of even the worst rechargeable batteries (30 Wh/kg).

[Next steps: quality-assure the numbers in the table, and do a real-world check against the actual weight and actual energy stored in real clock mainsprings.]


Unknown said...

When converting your values from Wh/kg to J/g the value for titanium seems very small. Is the equation for your specific energy applicable to compression springs? Or does this only apply to wound-up springs (e.g. in clocks)

Abhishek Dogra said...

Thanks a lot sir

Buzun said...

I'm doing something wrong, can you tell me what?

0.5 (τmax)2 / ( Y ρ ) for steel A36 => (0.5 * 250 * 250) / (200 * 8000) => 31250 / 1600000 = ~0,0196 and you've got 0.005.

Where did I go wrong?

Unknown said...

Any Answers to this question? I am not getting correct answers either. Units converted to pascals. Still not working.

David Spector said...

It is always fascinating to me that an author would take the time to write a good post, but then ignore all comments. During my career as a software engineer, I actually held end-user feedback (such as bug fixing) as being my top priority, over design and problem solving. I found that putting people first is the best policy in life. This does more than anything else to maintain quality and knowledge.

sriram said...

Commentors here don't seem to be aware the Prof. MacKay passed away in 2016. This post was written from his hospice bed.

David Spector said...

I apologize for my critical posting. I had no idea. But his message was posted May, 2014, which is two years before his death.

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