Determining energy from spring arrangements (parallel&series)

[R_Rose]

I'm trying to figure out how to calculate the difference between applying different extremes in spring arraingement for tension applications parallel'd, series and mix of them.I have some springs which are about 18” long and stretch to 30”, so they stretch 12”. If 8 springs are available and the use is set up similar to how a cross bow launches something – the springs contract, pulling a cable that connects to a carriage/carrier, launching something (there is probably a pulley to change direction by 45-90 degrees). I'm trying to see how 3 different configurations would work.

Option
A: 8 in parallel – total stretch 12”
B: 4 in series – 2 parallel sets – total stretch 48”
C: 8 in series – total stretch 96”

Examples of use: Launching drone airplane (say 15bs), shoot spear of some kind (3lbs), Launch steel ball – shot put (12lbs)

I saw that the springs are rated at 4,300 N/m which seems pretty high to me, and I'm not sure if it is accurate or not.

I can obviously see that Option A will accelerate quickly over short distance while C will be slower over longer distance. What I'm interested in is the relationship between the two, and what happens in between. Also, when changing the weight of the item from 3-15lbs, how does this effect things (obviously the heavier will accelerate slower and not travel as far) is the travel distance a linear relationship.

I'm also wondering how using something like a block and tackle to increase travel distance of the carriage – if I were to use 8 in parallel and 4 pulleys on the tackle (so energy should be the same as option B – is that correct??)

I hope I've made my questions clear and you can see what I am trying to get at with this. If you need something made more clear, please let me know. Thanks for any help you can provide on this!

 

[sophiecentaur]

I think a few basics about stretching springs would help you on your way with this.
This hyperphysics link is brief and to the point. It will also tell you the energy stored in a spring. (SI units, I'm afraid). The Spring Constant k is the stiffness.
For different arrangements of (identical) springs you can work on the principles. For n springs in series, they have the same tension and n times the extension of a single spring under the same tension and, for n springs in parallel, they will have 1/n times the extension for the same load as for a single spring or the total load needs to be n times that for a single spring.
You start with your initial spring arrangement and values of, say tension and k and that will tell you your extension and the stored Energy.
You can convert the stored energy (Potential Energy) and equate it to the Kinetic Energy of the projectile mv²/2 and that will tell you its final speed. The acceleration will be greatest when the tension is greatest (Force = Mass X Acceleration - from Newton's 2nd Law)
There is another issue. If you are contemplating a block and tackle to increase the speed, the efficiency of the pulley system will be very relevant and it may be better to avoid that arrangement. This is why they used to use a long arm on a Ballista, which gives you distance multiplication without much Energy loss. (Except of course the wasted energy in accelerating the lever itself! You can never win) Modern blocks may be a lot better than what The Romans had available.
If all this is for a practical project then you should read around the development of Throwing Weapons. The problems were identified and solved many hundreds of years ago.

 

[CWatters]

if I were to use 8 in parallel and 4 pulleys on the tackle (so energy should be the same as option B – is that correct??)

If all springs are stretched to their maximum length then the stored in all three options would be the same. The energy stored in 8 batteries doesn't change if they are connected in series or parallel or any combination.

If the spring constant for one spring is k then the spring constant for each option is:

A: 8k
B: k/2
C: k/8

See also https://en.wikipedia.org/wiki/Series_and_parallel_springs

The energy stored in a is 0.5kx² where k is the spring constant and x the extension. If x_max is the max extension of one spring (eg 12")then the energy stored for each option becomes..

A: 0.5(8k)(x_max)² = 4kx_max²
B: 0.5(k/2)(4x_max)² = 4kx_max²
C: 0.5(k/8)(8x_max)² = 4kx_max²

So if all the individual springs are stretched the same then all three combinations store the same energy. So with some assumptions it would be reasonable to expect all three to launch the projectile with the same velocity.

The force needed to stretch each spring is kx. So the force produced at maximum extension for the three options would be..

A: (8k)x_max = 8kx_max
B: (k/2)(4x_max) = 2kx_max
C: (k/8)(8x_max) = kx_max

So the block and tackle would have to be 8 times stronger for option A than C. That's what you would expect.

Newton says F=ma or a =F/m so although the launch speed should be similar the acceleration will be different for each option. Again that's what you would expect because each has a different distance over which to get the projectile up to the same launch velocity.

Think I have all those numbers correct.