## Total Airship Lift Capacity

I have now calculated the airship’s lifting capacity for all 4 of it’s gas bags in previous posts. There will be one in the nose, two in the middle, and one in the tail. So here is what I have calculated just to recap.

**Note**: There will be almost no space between the gas bags within the finished airship. I’ve added a couple inches between them in the diagram (right) for illustration only.

Volume: 2.4469 ft^{3}

Lift Capacity: **69 grams**

Middle (Fore)

Volume: 5.4002 ft^{3}

Lift Capacity: **152.2856 grams**

Middle (Aft)

Volume: 5.4002 ft^{3}

Lift Capacity: **152.2856 grams**

Volume: 4.3412 ft^{3}

Lift Capacity: **122.42 grams**

So it looks like the airship will have a **total gas volume of 17.5885 ft ^{3}** and a

**lift capacity of 495.9912 grams**. Google says there are around 453.5924 grams in a pound. So the airship’s helium volume should,

*hopefully*, be able to

**lift around 1.09 pounds**. At this point it seems to be more than enough, but we’ll see as I work through the rest of the frame and equipment weight calculations..

## Calculating Gas Bag Volume / Tail

I decided it would be a good idea to calculate the lift capacity for the whole ship before continuing the build work, and have now moved on to the tail gas bag. Once again, there will only be a single gas bag in the tail, but I have drawn it in ring segments (Fig. 1) to make calculating the volume easier. We will calculate the volume of each ring segment (Fig. 2), then add them all together, to get the total volume of the aft (tail) gas bag.

*Since Sketchup will not draw smaller increments than 1/16″, so once again I have again rounded up 1/32″ (radius) in a couple of places. The calculations should still be fairly accurate though.*

We will be using the same equation we used to calculate the volume of the nose gas bag to calculate the volume of the tail segments. There is also an online calculator that can also make this easier, but for now we will work the equations manually to illustrate the process.

V = (pi * h / 12)(d^{2} + db + b^{2})

Each segment of the aft (tail) gas bag, with the exception of the tail cone at the end, is a “frustum of a cone” shape (*a cone with the top cut off*). And each segment is separated by the frame ring where the bevel/angle happens.

We will work the calculations from left (largest) to right (smallest) *as pictured above*, denoting each as section **A**, **B**, **C**, **D**, **E** and **F** (cone). *I am again using only 4 decimal points for everything, but if you notice an error in my math (it happens) don’t be shy and let me know!*

Section A

V = (3.1459 * 6 / 12)(21.625^{2} + 21.625 * 22.25 + 22.25^{2})

V = 1.570795 * 1,443.8594 = 2,268.0071 in^{3} (*or* **1.3125 ft ^{3}**)

Section B

V = (3.1459 * 6 / 12)(19.875^{2} + 19.875 * 21.625 + 21.625^{2})

V = 1.570795 * 1,292.4531 = 2,030.1789 in^{3} (*or* **1.1749 ft ^{3}**)

Section C

V = (3.1459 * 6 / 12)(17.0^{2} + 17.0 * 19.875 + 19.875^{2})

V = 1.570795 * 1,021.8906 = 1,605.1806 in^{3} (*or* **0.9289 ft ^{3}**)

Section D

V = (3.1459 * 6 / 12)(12.875^{2} + 12.875 * 17.0 + 17.0^{2})

V = 1.570795 * 673.6406 = 1,058.1513 in^{3} (*or* **0.6124 ft ^{3}**)

Section E

V = (3.1459 * 6 / 12)(6.75^{2} + 6.75 * 12.875 + 12.875^{2})

V = 1.570795 * 298.2344 = 468.4651 in^{3} (*or* **0.2711 ft ^{3}**)

Section F (Tail cone)

V = ((pi * r^{2}) * h) / 3

V = ((3.14159 * 3.375^{2}) * 6.0) / 3 = 71.5693 in^{3} (*or* **0.0414 ft ^{3}**)

So it looks like the aft (tail) gas bag should have a **total volume** of approximately **4.3412 ft ^{3}**. Since a cubic foot of helium can lift around 28.2 grams, this should give the aft gas bag a total

**lift capacity**of around

**122.42 grams**.

I have not yet figured out what the frame, and tail fins, will weigh though. This is probably enough to lift them though, since a good portion of the tail will be built from 1/16″ balsa sticks. But if not, there is some excess lift capacity in the two middle sections to compensate for it. I will finish calculating the whole frame weight in a future update just to be sure.

## Calculating Gas Bag Volume / Nose

I decided it would be a good idea to calculate the lift capacity for the whole ship before continuing the build work, starting with the forward gas bag in the nose. There will only be one single gas bag in the nose, but I have drawn it in ring segments (Fig. 1) to make calculating the volume easier. We will first calculate the volume of each ring section (Fig. 2), then add them all together, to get the volume of the forward (nose) gas bag.

*Since Sketchup will not draw smaller increments than 1/16″ I have rounded up 1/32″ (radius) in a couple of places. There is enough space in between the rings for the gas bag to expand a little that the calculated volume should still be okay.*

Like many, I’m no expert with geometry. And some formulas I found are a little too math nerd for me. But I found a very useful online calculator that makes it easy to calculate the volume and surface area of the frustum of a cone. The volume calculation is based on the following formula:

V = (pi * h / 12)(d^{2} + db + b^{2})

Each segment of the forward gas bag, with the exception of the cone at the end, is a “frustum of a cone” shape (*a cone with the top cut off*). And each segment is separated by the frame ring where the bevel/angle happens.

We will work the calculations from left (largest) to right (smallest) *as pictured above*, denoting each as section **A**, **B**, **C** and **D** (cone). *I am only using up to 4 decimal points for everything, but if you notice an error in my math (it happens) don’t be shy and let me know!*

Section A

V = (3.14159 * 6 / 12)(20.75^{2} + 20.75 * 22.25 + 22.25^{2})

V = 1.570795 * 1,387.3125 = 2,179.1835 in^{3} (*or* **1.2611 ft ^{3}**)

Section B

V = (3.14159 * 6 / 12)(18.0^{2} + 18.0 * 20.75 + 20.75^{2})

V = 1.570795 * 793.3125 = 1,246.1313 in^{3} (*or* **0.7211 ft ^{3}**)

Section C

V = (3.14159 * 6 / 12)(6.875^{2} + 6.875 * 18.0 + 18.0^{2})

V = 1.570795 * 495.0156 = 777.568 in^{3} (*or* **0.445 ft ^{3}**)

Section D (Nose cone)

V = ((pi * r^{2}) * h) / 3

V = ((3.14159 * 3.4375^{2}) * 2.75) / 3 = 34.0288 in^{3} (*or* **0.0197 ft ^{3}**)

So from what I gather so far it looks like the forward (nose) gas bag should have a **total volume** of approximately **2.4469 ft ^{3}**. Since a cubic foot of helium can lift around 28.2 grams, this should give the forward gas bag a total

**lift capacity**of around

**69 grams**.

This may not be enough to lift the frame of the nose (*I have not calculated that yet*), but there will be some excess lifting capacity left over in the middle sections that can compensate for this, if needed. I will most likely also build most of the nose section from smaller 1/16″ balsa sticks too, so that should help keep the frame weight down.

## Gas Bag Volume/Lift and Weight Calculations

I was initially trying to decide on a diameter for my airship. At first I was working some math on both a 3′ diameter and a 2′ diameter. Then I remembered that I live in an apartment, and I will need to be able to fit this thing through a normal single door.. So 2 foot diameter it is!

To start I created a **2′ diameter ring**, with **16 sides**, using Sketchup. The program makes it easy to then draw a circle (arc) from the center out until it meets up with the inside of the rings, and then I can get the area of the circle too with a right-click menu choice. *But this ended up wrong since it’s not measuring a circle, but a polygon..* So I used 1′ 10 1/4″ for the diameter (rounded up a bit).

Most folks that graduated high school learned how to calculate the volume of a cylinder. But if high school was a couple of decades ago there are online tools to do it for us! Basically, the *area of the base* x the *height* of the cylinder = *volume*. So..

Area = πr^{2}

3.14159 x 11.125^{2} (11 1/8″ radius) = 388.8208 in^{3}

..and we know that there are 144 (12 x 12) inches in a square foot, so..

388.8208 / 144 = 2.7001 in^{2}

So we know that each 1 foot length of this cylinder has a volume of 2.7001 ft^{3}, and our test section is 2 feet long, so the **gas bag** should have a **volume** of **5.4002 ft ^{3}**.

Anyway.. Now we need to find out if this gas volume can lift the frame as we’ve designed it. So we will add up the weight of the construction materials (minus glue) to see if this design might actually work.

*I may post some data on the calculations with carbon fiber rods later, but at this point I’m pretty sure I’ll be building the test section with balsa wood sticks. So for now we’ll focus on the weight of a balsa wood frame.*

The **outer part of the rings**, and the **longitudinal stringers**, will use **1/8″ balsa sticks**. And the **inner bracing** of the rings will use **1/16″ balsa sticks**. I weighed a 36″ section of the 1/8 balsa and it weighed 2 grams. So a 36″ section of the 1/16″ should weigh 0.5 grams.

This **2′ test section** of the frame will use 63′ 3″ of the 1/8″ balsa and 124′ 6″ of the 1/16″ balsa (probably a bit less, since I’m not accounting for bevel cuts). The weight of the 1/8″ sticks should be about 42.1667 grams, and the weight of the 1/16″ sticks should be about 10.2083 grams. So the **total weight** of the **balsa wood frame** should be around **52.395 grams**.

We know that **1 cubic foot of helium can lift about 28.2 grams**. So the **5.4002 ft ^{3}** our

**gas bag**holds should be able to

**lift 152.2856 grams**(about 5.4 ounces). So we’re doing well so far since the frame only weighs 52.396 grams! Stay tuned and we’ll factor in the weight of the exterior covering and the gas bag materials next.