Fine Tuning ThunderStruck’s 45km Free Fall
By Robert Brand and Todd Hampson
This post will examine a drop from 45km. Todd has done an amazing job on this interactive Excel spreadsheet. We can change a few variable and see the change effects. It has done an amazing job in letting us see what changes produce the greatest “bang for the buck”.
The first thing was changing the diameter of the craft creates a great difference in drag. We have decided that we need to make the fuselage 300mm in diameter (down from 600) as it have a huge effect on when the craft stops accelerating. It adds Mach 0.3 to the top speed. from a 45km drop. We also noticed that if we get the craft the right dimensions and mass, the need to get the balloon to 45km is reduced. We can still break the sound barrier with a release from 40km altitude. At 45km we get a top speed of Mach 1.54 and at 40km we get Mach 1.36. This is also with a mass of 15kg rather than 10kg as we previously had though would be sufficient. We do not want to release much lower as things change rapidly with the thicker air.
Above is the top part of Todd’s spreadsheet, the coloured cells to the left allow different inputs and the cells on the right are the snap summary. The model that we have made only just got us over Mach 1 with little to spare. Changing the diameter and elongating the nose (a smaller 1/2 vertex angle of the cone) made a huge difference and making the mass 15km means a huge leeway. As mentioned on TV recently, we are aiming for Mach 1.5 and now we have the maths to prove that we can reach that speed. One interesting aspect of the reaching Mach 1.5 is that the deceleration by the thickening atmosphere is about 1.3G. Barely more than standing on the ground. It is a really gentle load and it is mainly on the nose cone of the craft. The wing and tail assembly will keep the craft oriented in the denser air and we will rely on the ballast in the front of the craft to keep it nose down. The ballast is likely to be antifreeze and we can shift it or eject it for a more stable and slower level flight.
We hope to have the fully interactive spreadsheet available on the site for those interested, but until then let’s have a snapshot of the curves that count. That is a free fall from 45km.
At sea level, mach 1 is about 340m/s. I say “about” because air pressure has little to do with the speed of sound. It is mainly air temperature. From the graph we should reach 530m/s and that is Mach 1.56.
Before we streamlined the craft to punch through the thickening atmosphere, the wider bodied version of ThunderStruck slowed down really fast and took some stronger G force on the nose (mainly). The version 2 craft slows almost at the same rate that it accelerates. This gives a very gentle change as can be seen below.
From the graph above it is clear that at 45km, as the speed increases, the air resistance has a greater effect. At that height air density is about 0.025 10-1 kg/m3 compared to air density at sea level which is about 12.25 10-1 kg/m3 (plus or minus about 5%).
This means that our air density at 45km as a percentage of air density at sea level is about 0.284% that of sea level and it increases as we go lower. The effects also increase with ThunderStruck’s speed as the drag has a greater effect with both speed and increasing density.
With the calculated drag of the craft, we find that all acceleration stops at 26 km and as we fall into denser atmosphere, we begin to slow. The graph above is calculated in metres per second per second (known as m/s/s or m/s2) and that can be directly converted in to G force. Since 1 g = 9.80665 m/s2 a simple rule of thumb conversion to remember is 10m/s/s = 1G.
Now for many the next part of this may be hard to grasp, but at free fall at 45km we have what is loosely termed 0G where, if we were in a craft also falling at the same rate, we would float inside the craft. Once we reach terminal velocity at 26km altitude ( I will ignore the lag in deceleration here), we have 1G acting on the craft. If we were inside that craft we could walk around the interior and feel the same as on the earth’s surface (again small variations in gravity, etc excluded). A skydiver that has reached terminal velocity has the air flow stabilising his speed and that air flow has a force of 1G on his body. G force real is only noticeable when there is change – ie a change in direction or acceleration or deceleration.
The “vomit comet” aircraft that simulates zero G does so by moving steadily in a straight line while accelerating towards the ground at 9.8m/s2. If they just dipped the nose and began that arc, but stopped accelerating towards the ground we would all feel an initial 0Gs but then would be back on the floor when the rate of change returned to zero and we would be back at 1G. So, with ThunderStruck, it is the rate of change that determines G force and at 26km altitude, the G force is 1G on the overall craft, but since the greatest drag is caused by the nose, the 1G force is felt here. Other parts of the craft would be happy to continue accelerating! So at 26km, the structural form of the craft must allow the nose to hold the craft by the nose vertically – good to know, but it does not stop there.
The craft continues to slow and decelerate with the denser air and we have to slow way more. That now takes us into the realm of more than 1G. In fact at 17km we experience the greatest rate of deceleration or change and that is an additional 1.25G for a total of 2.25G on the nose of the vertical craft. That is the base amount of structural integrity we will need in the nose assembly. If the craft weighs 15kg, then the nose assembly has to support 33.75 and then an amount that we required to ensure it is strong enough. My design had better look to supporting 50kg on the nose when the craft is stood vertically at least.
It seems that what you gain, you have to give back. The higher the speed and the longer the period in low Gs, then the the higher the Gs or the longer in negative Gs you need to complete the flight back to a complete vertical stop. I have not analyses the areas on either side of the 0m/s/s on the chart above, but I would not be surprised if they where equal. As we say here – swings and roundabouts. What you gain on the swings, you will lose on the roundabouts.
Mathematics is a wonderful tool for designers. From a few simple facts in a spreadsheet, we have calculated the speed at all points in the flight (vertical dive perspective) and also the internal forces on the craft at many points. ie the winglet tips will be a point of high drag so they will need to handle more than 1G vertically. The same with other leading edges and that also goes for surfaces affected by shock waves. All of which can be determined by design and software. You don’t have to be a maths genius, but you do need to know maths enough to ensure that you can use them in day to day work. Unless you visualise what is happening, you will have an unhealthy reliance on software for everything you do. That often denies the genius of innovation. It is also why a novice can invent something a seasoned engineer fails to see.
By manipulating the graph by fine tuning the inputs we found that our craft accelerated longer or you could say “the rate of deceleration was slower” by:
- Making the nose cone pointier
- Making the fuselage (and the nose cone) a a smaller diameter
- Increasing the weight of the vehicle
In fact with the new design we have found that we can still break the sound barrier at a starting altitude of 40km. that is our plan B if the weather or winds in the atmosphere go against us. ie, we can launch early if the winds are taking our balloon out of range of our communications systems.
So what does our new design look like?
This is an early look as there are a few bits at the rear that still need adjusting.
We also took the opportunity to correct a few other aspects of the craft:
- Bigger wing Area with a larger area ahead of the main wing
- Longer spikes on the winglets (the winglets are not as high due to the smaller fuselage). This is to move the supersonic shock waves away from the control surfaces on the rear of the wing.
- Twin rudders trailing the craft (there are some wing tabs in the drawing that need to be removed.
- A tapered tail to stop high drag behind the craft (we also need to remove some wing tags in the model above.
- Tapered rudders on the bottom to stop it hitting the ground on landing (not shown)
- Tapered rudders on the top for symmetry to ensure that it has little differential in forces on the craft to make it pull out of the dive.
So there you have the new design based on maths and simulations on a home computer. It seems that building a supersonic aircraft is child’s play as Jason (12) is jointly working on this design. This morning I asked him what G force is at work on a skydiver at terminal velocity and he confidently answered “1G”. Good one grasshopper. He then went on to clearly say that g force was related to change in acceleration (relating to a skydiver). I love it when he talks maths. He needs to know as he will be the remote control pilot for this Mach 1.5 aircraft.