Length Change with Force - Now Change the Temperature !

[rsterne]

I need to figure out how much a barrel or shroud will change in length (elastically) when the barrel is tensioned by placing the shroud in compression.... I have never done this before, so please help !

From what I think I understand, Young's Modulas E is the ratio of Stress / Strain, and the units are psi.... Steel has an E = 29 x 10^6 (close enough), Aluminum about 10 x 10^6, and Carbon Fibre Reinforced Plastic (CRP) about 22 x 10^6.... Strain is the length change in (in/in), ie the length change per inch of length, and for this example our length L will be 24".... Stress is also in psi, and is the applied force divided by the area of the stressed member, F/A.... For this example, our force F = 1000 lbs. and the Area A will be a variable.... We are going to use a 1/2" OD Steel barrel with a 1/4" bore, and two different shrouds, both 1" OD.... One is Aluminum with a 0.035" wall, and the other is CRP with a 0.063" wall.... Since E = Stress/Strain, then Strain = Stress/E.... Now let's see if I've got this right....

Steel Barrel:
Area = ((0.5^2)-(0.25^2)) x PI/4 = (0.25 - 0.0625) x 0.785 = 0.147 in^2
Force = 1000
Stress = 1000 / 0.147 = 6803 psi
Strain = Stress / E = 6803 / 29x10^6 = 0.00023 (in/in)
Elongation of the 24" long steel barrel under a tension load of 1000 lbs. is 0.00023 x 24 = 0.0056"

Aluminum Shroud:
Area = ((1.0^2)-(0.93^2)) x PI/4 = (1.0 - 0.865) x 0.785 = 0.106 in^2
Force = 1000
Stress = 1000 / 0.106 = 9434 psi
Strain = Stress / E = 9434 / 10x10^6 = 0.00094 (in/in)
Shortening of the 24" long aluminum shroud under a compressive load of 1000 lbs. is 0.00094 x 24 = 0.0236"

CRP Shroud:
Area = ((1.0^2)-(0.874^2)) x PI.4 = (1.0 - 0.764) x 0.785 = 0.185 in^2
Force = 1000
Stress = 1000 / 0.185 = 5405 psi
Strain = Stress / E = 5405 / 22x10^6 = 0.00025 (in/in)
Shortening of the 24" long CRP shroud under a compressive load of 1000 lbs. is 0.00025 x 24 = 0.0059"

So if I have understood the method, and done the math correctly, a 1/16" wall CF shroud and a 1/2" steel barrel would both change length by just under 0.006", while a 1/32" wall Aluminum shroud would compress about 4 times as much at 0.024".... Comments, corrections, confirmations?....

Bob

[QVTom]

Looks correct to me.  I know it's just an example but 0.035 is a pretty flimsy shroud regardless of the material.

Tom

[hutnicks]

...but the bellevilles  are going to have to factor in there somewhere.

[rsterne]

Thanks, Tom.... I actually did a buckling calculation on the .035 aluminum shroud, and it's good for something like 1500 lbs....

OK, so now that I have an idea how much the barrel grows and the shroud shrinks under 1000 lbs. of load, I can look at the thermal stability of a straight torqued barrel/shroud assembly with no Bellevilles.... As I understand it, if the materials are the same, no problem, they will both expand and contract at the same rate, keeping the same load over any (reasonable) range of temperatures.... Here are some Thermal Coefficients I found....

Aluminum: 13.0 PPM / *F
Steel: 6.7 PPM / *F
Stainless Steel: 9.2 PPM / *F
4140 CrMoly: 13.7 PPM / *F
Carbon Fibre RP: 1-2 PPM / *F

So, surprisingly , Aluminum and CrMoly are almost identical.... However, either of those for a shroud with a steel barrel, the shroud would expand nearly twice as much.... and CF expands hardly at all.... So, let's take our 24" barrel and shroud, and see what happens if we change the temperature by 50*F (ie freezing to 82*, or 50* to 100*F).... If we multiply 24 x 50 we get 1,200, and since the Thermal Coefficients above are in parts per million, we would get 1,200/1,000,000 = 0.0012" times the Coefficient above for the amount of expansion.... So for a 24" barrel or shroud, and a 50*F change, we get the following length changes....

Aluminum: 0.016"
Steel: 0.008"
Stainless Steel: 0.011"
CrMoly: 0.016"
Carbon Fibre: 0.002" (using the average value)

Let's apply those to the combinations above, starting with them torqued to 1000 lbs. load cold (rounding to the nearest 0.001")....

Steel Barrel and Aluminum Shroud:
Shroud grows 0.016" - 0.008" = 0.008" more than the barrel when they are heated 50*F (0.008" tighter)....
The original length change from torqueing to 1000 lbs. load was 0.006" + 0.024" = 0.030"....
The new length difference is 0.030" + 0.008" = 0.038"....
I think that means that the force has gone up from 1000 lbs. to (0.038/0.030)x1000 = 1267lbs....

Steel Barrel and Carbon Fiber Shroud:
Barrel grows 0.008" - 0.002" = 0.006" more than the shroud when they are heated 50*F (0.006" looser)....
The original length change from torqueing to 1000 lbs. load was 0.006" + 0.006" = 0.012"....
The new length difference is 0.012" - 0.006" = 0.006"....
I think that means that the force has gone down from 1000 lbs. to (0.006/0.012)x1000 = 500 lbs....

You can see where I'm going with this when I add the Bellevilles into the equation, right?.... Anybody want to comment, disagree, or agree with my reasoning so far?....

To be continued tomorrow.... If I'm wrong, please tell me now.... *LOL*....

Bob

[rogue909]

Thanks, Tom.... I actually did a buckling calculation on the .035 aluminum shroud, and it's good for something like 1500 lbs....
.......

Steel Barrel and Aluminum Shroud:
.......
I think that means that the force has gone up from 1000 lbs. to (0.038/0.030)x1000 = 1267lbs....

Like I said in the other post when there was a discussion regarding thermal expansion; I am but a nubling for mechanical engineering, on my final senior semester this spring. But it seems to me that the safety factor is a tad small there. My field experience has been rather limited and focused on "small chat" with engineers but its my understanding that you typically want to keep a safety of 2-2.5~ for most things and 5-7~ for structural. (minimal)

I'm still not entirely sure how much this will effect barrel harmonics.  Due you have a post discussing tensioning barrels and the waves that are propagated? I did a few searches and couldn't find anything. Perhaps I am using the search function wrong?

[PakProtector]

The bit I would consider is calling the barrel 4130; in the case of 'steel' TJ barrels this is certainly the case. With other barrels, who knows? So while you show thermal expansion for an Al shroud being comparable to a 4130 barrel...there is a substantial hit to stiffness based on the Al Young's Modulus. I am quite comfortable with a 20 million psi YM for CF...which is twice as stiff as Al of the same geometry.
cheers,
Douglas

also, on the change in tension due to delta T needs loading adjusted. If the cooling leaves the shroud under more compression, the barrel will stretch more, which means the shroud will be under less compression.

[rsterne]

Normally I would agree with you, Ben, about safety margins, but in this case if it fails you get a kink in the shroud as it collapses a few thou.... not life threatening in any way.... I just happen to have some 0.035" wall aluminum tubing to experiment with, so I used it for an example.... Also, at this point, we don't know how much load is required to tension the barrel.... If it fails, I'll switch to CF or a thicker wall.... There is a discussion in the PCP Gate on Adjustable Barrel Tensioning which deals with the concept and some baseline testing, I just didn't want to cause the interest there to glaze over with a bunch of math....

Douglas, in my example I actually used Steel, not CrMoly for the thermal coefficient (6.7 PPM/*F).... I thought I took the barrel stretching into account by adding the amount of stretch to the amount of compression and then adding the difference in the thermal expansion to that total.... then using the percentage TOTAL change to back-calculate (one could say backhandedly, if that's a word) the change in the tension/compression force.... If you have a better method, I'll listen....

Bob

[rsterne]

OK, so now we put some Belleville washers into the mix.... The ones I have chosen are 1/2" ID x 1" OD x 0.073" thick and 0.091" tall, so they have a maximum "travel" of 0.018", and it takes 1335 lbs. to drive them flat.... That gives them a spring rate of 74,167 lb/in. if used alone.... However, my plan is to stack three of them in series, which increases the travel to 0.054" and decreases the rate to 24,722 lb/in, or 25 lbs. per thou (close enough).... The reason I chose that combination is so that I can get 1 turn of adjustment on the nut, which is 20 TPI, within that 0.054" of travel.... A side benefit is the softer spring rate allows more thermal expansion change without as much force change, which is what I plan on looking at next.... Using the same methodology as above, I will add the compression distance for the 3 Belleville stack at 1000 lbs. of load, which is 0.040", to the above examples....

Steel Barrel and Aluminum Shroud with Bellevilles:
Shroud grows 0.016" - 0.008" = 0.008" more than the barrel when they are heated 50*F (0.008" tighter)....
The original length change from tightening to 1000 lbs. load was 0.006" + 0.024" +0.040" = 0.070"....
The new length difference is 0.070" + 0.008" = 0.078"....
I think that means that the force has gone up from 1000 lbs. to (0.078/0.070)x1000 = 1114lbs.... (1267 lbs. without Bellevilles)

Steel Barrel and Carbon Fiber Shroud with Bellevilles:
Barrel grows 0.008" - 0.002" = 0.006" more than the shroud when they are heated 50*F (0.006" looser)....
The original length change from tightening to 1000 lbs. load was 0.006" + 0.006" + 0.040" = 0.052"....
The new length difference is 0.052" - 0.006" = 0.046"....
I think that means that the force has gone down from 1000 lbs. to (0.046/0.052)x1000 = 885 lbs.... (500 lbs. without Bellevilles)

You will note that this is a drastic reduction in the CHANGE in the tension/compression forces with temperature, compared to without the Bellevilles, which is what I expected to see.... Note that it's not the actual numbers that I'm concerned with, as the confirmation of the effectiveness of using Bellevilles.... To me, it certainly seems worthwhile to include the Bellevilles into the system, particularly with the ease of adjustability (a whole turn) and repeatability (flats and points on the nut, or a vernier scale).... If you wanted to have even a smaller percentage change with temperature, you could use more Bellevilles to reduce the spring rate further, but the tensioning process would then cover more than 1 turn on the nut.... That may or may not be an issue, of course....

Something to note is that for the CF shroud, which doesn't expand much with temperature, if used in combination with a barrel that was CrMoly, which expands nearly twice as much as the steel barrel, a torqued combination without the Bellevilles could very nearly go loose at high temperatures.... It certainly could reduce the tension from a lot to so little that it might be ineffective in reducing barrel harmonics.... Introducing the Bellevilles into the system may be the solution to being able to even use that combination of materials.... On the other hand, using the same material for both barrel and shroud, or using an Aluminium shroud and CrMoly barrel (which have almost identical thermal expansion) should allow a torqued setup to work throughout a wide temperature range.... However, IMO it still lacks the adjustability and repeatability of the tensioning system using Bellevilles....

Bob

[QVTom]

I've used three different shroud materials of various thicknesses: 0.065 6063T8, 0.093 6061T6, 0.065 and 0.090 4140.  The 0.065 6063 was  the most flexible by far when applying hand pressure to the muzzle, the 0.093 4140 is clear winner with the 0.093 6061T6 not far behind.  The most interesting observation was the 4140 in both thicknesses what noticeably quieter than the aluminum.

0.035 seems so thin and may not add much stiffness.  I have 0.065 uni CF tube that I haven't tried yet.  I think it will perform well other than the possible thermal differences but is just  plain ugly and will need to be primed and painted which feels like work at this point. 

I have a Hardinge CNC lathe that has an unique method to add stiffness and stability to the base casting.  They cast a polymer filler into the casting called Hardcrete, this adds vibration dampening, rigidity and of course added weight.  I wonder if there is some type of open cell or filled polymer that isn't too heavy we could cast into our shrouds? 

Tom

[rsterne]

Pour-in-place high-density foam?....

I guess if I see a difference with my 7/8"OD x 0.035" wall aluminum shroud on a 7/16" Crosman barrel (with thinner 7/8" Bellevilles) it will pretty much prove the concept then?.... The other test I have planned is a 1" OD x 0.063" CF (twill over uni) shroud on a 1/2" CrMoly barrel....

Bob

[QVTom]

Pour-in-place high-density foam?....

I guess if I see a difference with my 7/8"OD x 0.035" wall aluminum shroud on a 7/16" Crosman barrel (with thinner 7/8" Bellevilles) it will pretty much prove the concept then?.... The other test I have planned is a 1" OD x 0.063" CF (twill over uni) shroud on a 1/2" CrMoly barrel....

Bob

Bob, you are much more deliberate in your experimentation and testing than I (huge compliment).  I'm sure you will see results with the .035 but I assume they will be small.  The contrast between al and cf should be extremely telling and hopefully shed light on the primary benefit; reduced resonance from tension or stiffness due to effective diameter. 

I think the bellevilles are a great idea and may outperform their added complexity.  Loss of shroud volume is a small price to pay for increased accuracy.

I'm not sure foam is  structural enough but there must be materials that are less dense than steel that can be used as fillers.  Any reasonable weight saving over a solid steel should be win although it's tough to beat the weight of air!

Tom

[rsterne]

Don't care about shroud volume in Canada, as anything that make the gun quieter (even by 1 dB in theory) is a Prohibited Weapon (yep, just like a machine gun! ).... I wonder if a non-functional 4140 shroud (or carbon) would be too quiet, even though the muzzle gasses never enter it.... *LOL*....

The foam suggestion was kind of tongue-in-cheek, but I understand your idea.... Stuffing the space with strips or washers made from Magic Erasers is probably the easiest (if you don't live in Canada).... I like the idea of saving weight, but if I was building a benchrest railgun I'd be using a 1.65" OD x 32" long LW blank in my chosen caliber and avoid all this nonsense....

What do you think of the idea of an intermediate (non-harmonically spaced, eg. 4/7 ratio) spacer(s) between the barrel and shroud to dampen vibrations and make the assembly work more like one unit?.... Any point, or just complicating matters?.... A set-screw to the barrel and O-ring against the shroud ID is the way I have gone in the past (non-tensioned).... No idea if it really WORKS of course.... *LOL*.... While on the subject, how about a barrel band supporting the shroud OD?....

Bob

[hutnicks]

I've used three different shroud materials of various thicknesses: 0.065 6063T8, 0.093 6061T6, 0.065 and 0.090 4140.  The 0.065 6063 was  the most flexible by far when applying hand pressure to the muzzle, the 0.093 4140 is clear winner with the 0.093 6061T6 not far behind.  The most interesting observation was the 4140 in both thicknesses what noticeably quieter than the aluminum.

0.035 seems so thin and may not add much stiffness.  I have 0.065 uni CF tube that I haven't tried yet.  I think it will perform well other than the possible thermal differences but is just  plain ugly and will need to be primed and painted which feels like work at this point. 

I have a Hardinge CNC lathe that has an unique method to add stiffness and stability to the base casting.  They cast a polymer filler into the casting called Hardcrete, this adds vibration dampening, rigidity and of course added weight.  I wonder if there is some type of open cell or filled polymer that isn't too heavy we could cast into our shrouds? 

Tom

http://www.synfoam.com/

[Bill G]

The bit I would consider is calling the barrel 4130; in the case of 'steel' TJ barrels this is certainly the case. With other barrels, who knows? So while you show thermal expansion for an Al shroud being comparable to a 4130 barrel...there is a substantial hit to stiffness based on the Al Young's Modulus. I am quite comfortable with a 20 million psi YM for CF...which is twice as stiff as Al of the same geometry.
cheers,
Douglas

also, on the change in tension due to delta T needs loading adjusted. If the cooling leaves the shroud under more compression, the barrel will stretch more, which means the shroud will be under less compression.

Very often  .22LR barrels are made of one of our favorite materials, 12L14.  Its ease of cutting and leaving minimal tool marks makes it great for lower psi and heat applications.  I would Imagine that goes even better with an airgun.  I've always wondered about another of mypersonal favorites, stress proof steel.  Cuts really nice too.     

[Bill G]

Bob, you probably already realize this but just incase.  If you want to reduce the change in tension, you could add bellville washers to the stack.  Wouldn't this allow the force to remain the same and be divided more?  So say that .010" causes 200lb change in force and doubling the amount of washers would allow a smaller resolution. so 100lb would be the new change over the same distance change.  That is assuming that the force is linear with change in deflection.  Bellevilles are real close to linear until you hit around 75%-80% of deflection. 

I may be misunderstanding what it is that you are trying to achieve.  If I understand correctly, you want to have the tension force change as small as possible.

Bill   

[rsterne]

Mentioned that above....

If you wanted to have even a smaller percentage change with temperature, you could use more Bellevilles to reduce the spring rate further, but the tensioning process would then cover more than 1 turn on the nut.... That may or may not be an issue, of course....

Tom found that past a certain point little was gained by further tensioning.... My hope is to simply operate beyond that tension at all times....

Bob

[Bill G]

I missed that.  Must of been skiming too fast.  I could use another couple of hours in each day .  Anybody have some for sale?

[PakProtector]

I can see the wish to keep tension constant at what ever proves optimum. However, as the barrel tries to wag back and forth, the Belleville washer stack will still try to keep the loading constant...even as the shroud tries to rock up onto a 'corner'...and the solid mount would drastically raise the tension at that point.
cheers,
Douglas

[rsterne]

Exactly the reason I argued against putting the Belleville stack at the back end against the breech in another thread.... I guess which is better, a torqued setup or the Bellevilles will depend on how rigid the shroud is (eg. Tom's comments about a 0.093" wall CrMoly tube being stiffer).... I can see the torqued setup working OK if you use a torgue wrench to set it up each time you remove and replace it, and IF the materials for the barrel and shroud have the same coefficient of expansion.... Using a CF shroud on a CrMoly barrel, not so much, unless you are into checking the torque as the temperature changes, IMO.... Everything is a compromise, right?....

Bob

[QVTom]

http://www.synfoam.com/

Some very interesting products.  Good link!