# Calculation of rate of climb (vertical velocity)

Hello my friends,

At the moment, I'm doing an estimation and analysis of flight performance for UAV (unmanned aerial vehicle). During climbing flight analysis, I have a problem regarding calculation of rate of climb (vertical velocity), my calculation value of rate of climb seem weird, the vertical velocity (RC) bigger than velocity vector. If my rate of climb is not correct, so I have faced also difficulty to find absolute and service ceiling. I hope someone can help me to solve this problem quickly, I have already calculate it many time but the result still same. I would like to attach xls file (example of analysis table), so I really want a help from all of you here. I don't know what's the problem in my analysis. Please help me ..... If all of you don't understand, ask me for explanation.

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• analysis table.xls
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Up front I must say that wading through other person's spreadsheet is quite difficult, as spreadsheets are just not reading-worthy. A better presentation would be made using something that can actually show formulas and comments in a readily understandable way, e.g. Matlab, with a sample evaluation for one input vector. Luckily, here you have quite a simple computation, and I needed a brief diversion, so...

The problem is that your aircraft is massively overpowered. In such a scenario, climbing velocity cannot be computed by the $(P_{avail} - P_{req}) / W$ formula -- this is an approximation that holds for limited thrust/power-to-weight designs.

On the other hand, while I don't know much about small IC engines, it seems unlikely to me that a 4 kW engine could weigh so little that the gross takeoff weight of the whole UAV is only 20 kg.

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Chusslove Illich (Часлав Илић)

Thanks my friend for your opinion. Actually, I just want to show the results of calculation using spreadsheet based on roskam method, so my friends in this forum could analyze. Can you explain to me about this statement

(a) climbing velocity (rate of climb) cannot be computed by the formula -- this is an approximation that holds for limited thrust/power-to-weight designs.

(b) it seems unlikely to me that a 4 kW engine could weigh so little that the gross takeoff weight of the whole UAV is only 20 kg.

Do you have any suggestion, what equations are suitable to solve this problem. It just not rate of climb problem, I'm also facing with the problems of service ceiling, absolute ceiling, range and endurance calculation. From my point of view, the engine is the main problem. What information do you need, so it will allow you to solve the problems?

caslav.ilic said:
[...] it seems unlikely to me that a 4 kW engine could weigh so little that the gross takeoff weight of the whole UAV is only 20 kg.

Well, after googling a bit, you can forget about this statement of mine. I thought that the engine wouldn't have power-to-weight higher than 1 kW/kg, but apparently there exist ~5 kW engines with >2 kW/kg PtoW.

The question is still whether you need such a powerfull engine. But that's for you (or someone else) to clarify, as I don't know what special requirements a ~20 kg UAV should have.

If you do need such an engine, then...

amirshah said:
[...] based on roskam method [...] an approximation that holds for limited thrust/power-to-weight designs.

I will assume that by "Roskam method" you refer to e.g. eq. 9.12 in Roskam's "Airplane Aerodynamics and Performance". In that case, observe that 9.12 is usable only if the conditions 9.5 are satisfied; most importantly, the flight path angle $\gamma$ should be small (say < 15 deg).

If you then refer to eqs. 9.3 and 9.4 (which preceede assumptions 9.5), in them you neglect thrust-to-body axis angle $\phi_{T}$ and acceleration $dV/dt$, rearrange them so that weight term is on the left and everything else on the right, and finally divide 9.3 by 9.4, you end up with:
$$\tan \gamma = \frac{T \cos\alpha - D}{T \sin\alpha + L}$$
or when multiplied by velocity to have it in power-formulation:
$$\tan \gamma = \frac{P_{av} \cos\alpha - P_{reqd}}{P_{av} \sin\alpha + LV}$$
From this you can see that when $P_{av} >> P_{reqd}$ and even ($P_{av} \approx LV$ (as in your case), the angle $\gamma$ will not be small. Thus, 9.5 is no longer applicable.

Do you have any suggestion, what equations are suitable to solve this problem. [...] What information do you need, so it will allow you to solve the problems?

No special equations are needed, the 9.3 and 9.4 and the usual aerodynamic force expressions ($X = 0.5 \rho V^2 C_X A$) will suffice.

What is needed is adept handling of those expressions. For example, for a given AoA $\alpha$, the remaining unknowns in 9.3 and 9.4 are the velocity $V$ and the flight path angle $\gamma$; but, the equations are non-linear, and have to be solved iteratively for these quantities. You could assume $\gamma_1 \approx 0$ and compute $V_1$, then use $V_1$ to compute $\gamma_2$, then use $\gamma_2$ to compute $V_2$, etc. until at one point $\gamma_k$ and $V_k$ pretty much stop changing. Then your climbing velocity for the given AoA will be simply the final $V_k \sin\gamma_k$.

I would like to say thank you for your help. I really confused about

[that the engine wouldn't have power-to-weight higher than 1 kW/kg, but apparently there exist ~5 kW engines with >2 kW/kg PtoW]

Yes, I'm refer to eq. 9.12 in Roskam's "Airplane Aerodynamics and Performance". I have re-read Roskam's book and observe that your ideas and comments are true. Equation 9.12 is good only if the conditions 9.5 are satisfied, the more steepest the climb is the worst this approximation will be, so if our UAV has a big specific excess power, it will climb quite steepy and the results can be not so good to use that equation. The climbing analysis at steep angles I need to use eqns. (9.59), (9.61), (9.62) and (9.64) to solve the problem. I will try these steps.

From your comments, I beginning to think that the main problem is the engine has massive power compare to weight.

amirshah said:

Pay no attention. I merely thought that the given engine would be too heavy to put in a UAV of the given size, but then took a better look and found out this is not the case.

The climbing analysis at steep angles I need to use eqns. (9.59), (9.61), (9.62) and (9.64) to solve the problem. I will try these steps.

Yes, these ought to work too. The difference to the procedure I stated above is that you fix the velocity instead of the angle of attack, and then iterate to find the flight path angle. This means that once you have it, you should also check whether the implied angle of attack is admissible (you compute it inversely from the lift curve).

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Chusslove Illich (Часлав Илић)