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Calculation tools
for ornithopter models

cover picture of the handbook 
'How ornithopters fly'
Cover picture of the handbook

There is no common calculation method for flapping wings. In every source of information different methods are being described.

All Orni calculation tools given below are based on the calculation method specified in the handbook How Ornithopters Fly. Particularly, the power and the twisting of the wing at stationary flight situations can determinate with it.

Furthermore, the calculation method permits an approximate quantitative specification of dynamics and aerodynamics of profiled flapping wings, at least. Primarily, the numerical comparison of various factors influencing of flapping wings.

Profiled flapping wings and quasi-stationary flow conditions are presumed. Therefore, calculations only lead to useful results for a fast forward flight with a relatively low flapping frequency (large birds, flying with lift).

The mathematical models are only executable with the respective software applications.

Orni 1

Simple calculation tool
to describe the wing twisting

The equations of the handbook are used in a small MS-EXCEL calculation tool:

Download Orni 1 (format xls, with a short instruction in English rel. 3.4, 0.2 MB)

Download Orni 1 (format xlsx, with a short instruction in English rel. 3.4, 0.1 MB)

This tool primarily allows the calculation of wing twisting at given wing data.

Force progression in principle
Force progression
in principle
force calculation scheme for flapping wings
Force calculation scheme for flapping wings

The additional calculation of data of the flying model is highly simplified. The detailed calculation of the force progression during a flapping cycle has been replaced by assuming a sinusoidal progression of the force. Thereby coupling of stroke motion and twisting remains unconsidered. Only the reference line (as gliding flight) and the amplitude of the sine half-wave (in the center of each wing stroke) are calculated. With that, the area under the curve progression or the size of a force is calculable.

Orni 2

Calculation method for ornithopter models

The basic calculation method of the handbook How Ornithopters Fly is reconstructed step by step. Many of the itemised equations are generated with accompanying text, diagrams and pictures.

The abidance of the operational range of the lift coefficient of the used airfoil is automatically warranted in the calculation tool. Also, the setting of balance of forces during flight happens in a simple way. Therefore, one can certainly achieve the flight data of the chosen model configuration.

For the construction of a flapping wing model the calculation method describes various force and moment developments at the wing. Furthermore, the required power and - above all - the wing twisting along the span are being calculated.

The calculation tool was written (in German) with the software Mathcad.
Because of the volume of data and pages it requires a little familiarisation.


Web Page
for taking a look

PDF file
for printing

for Mathcad-user

rel. 3.3

1.7 MB
317 KB
1.8 MB

Airfoil CLARK-Y
rel. 1.2

270 KB

56 KB

257 KB

Airfoil SD7062-PT

320 KB
60 KB

Crank drive

450 KB
110 KB

Orni 3

Mathematical model for ornithopters

This tool is based on the calculation tool Orni 2. Here, gliding and power flights of a flapping wing model can be analysed singly and in series.

To calculate an ornithopter model it is specified by diverse input parameters at first. One of these is defined as an area variable. In the chosen data area one then receives the data of flight and drive, while automatically keeping the data of the airfoil and the balance of forces.

Since there are only a few specifications included, the calculation tool Orni 2 beforehand should be known.

The calculation tool was written with the software Mathcad (in German).


Web Page
for taking a look

PDF file
for printing

for Mathcad-user

rel. 1.3

665 KB
130 KB
667 KB
(use airfoil programs above-mentioned)

The here listed calculation tools are too understood as open source software. Therefore, everybody is enabled to take an insight look in the calculation method. Everyone has the permission to relay, vary and particularly to correct and improve this source code.

Comparison of the transport performance of a swan
with flying models

The flight performance of a mute swan is compared with the performances of a propeller and a flapping wing model.

Download comparison.xlsx   (Excel table, 28 kB)

The mute swan consume the most energy. Particularly, this is due to the small efficiency of the muscles. Instead, the swan has the advantage that his body fat provides a relatively high density of energy storage. In the calculation tools flapping wing models are a little bit more effective than propeller models.

Also the flapping wing model of the handbook is compared with an according propeller model.