Introduction One of the problems that arises when building a model aircraft is the correct placement of the Center of Gravity. When assembling an ARTF or scratch building from a Plan the problem does not arise, as the designer should give full details on the plan or in the kit. However, when you are building to your own design you may need to work out where the C of G is located. This is not a problem with a "normal" model, but what about Biplanes, a Beech Staggerwing, Delta's, Canards & other odd layouts. Full size designers have powerful computers and wind tunnels but we must get it right for that first flight. Once in the air you are fully committed and an error will almost certainly cause a crash or at best a very twitchy flight. There is little point going into computational details here as there are one or two good programs on the Internet that will do most of the work for you. You will find links below to the ones I have located as they are quite hard to find and suggest you try these out, they all give more or less the correct answer. In the good old free flight days, test glides were the norm, trimming out a model into long grass until the model flew straight and level just off the stall. Test glides of heavy fast radio models are not possible so we need to get the C of G correct for that first flight. If in doubt use the old rule of thumb "1/4 of the wing chord back" This is generally not far out. See also the "model of a model" method.
Calculation Problems There are however a couple of problems. Most of the calculations involve an element of guesswork so the final result can only be at best described as a very accurate guess. For example tailplane efficiency varies between 30 and 100% and you need to make an educated guess as to the value you use in your calculations. A tailplane close to the wings trailing edge and in the wake vortex will come out as low as 30%, a "normal" location 60% whilst a canard (foreplane) is in the 95-100% range as it operates in "clean" air. A high set "Tee" tail will be closer to 90%. Do not bother with lifting tailplanes. A flat plate or thin symmetrical type is just as efficient. Secondly the C of G needs to be in front of the Neutral Point, but how far? Again a degree of intelligent guess work is required. The accepted figure is between 5 & 20%, 15% is a good compromised for first flights. See note below. Once you have a model that flies, at least well enough to land in one piece, you can then adjust the C of G based on the results of the first flight. Some links are given below where you can find Nomograms etc. to do most of the calculations for you and I will add others as I find them.
Fly by Wire Fighters Variously known as CCV (Control Configured Vehicles) or ACT (Active Control Technology) these aircraft are designed to be unstable and only fly under the full control of high speed computers, with minimum input from the pilot, where the roll and pitch sensors input to the control surfaces at 100 x per second. Not quite as fast as our 2.4GHz ! Some very successful large turbine models of full size CCV aircraft have been built and fly very well. The only difference being in the location of the C of G. CCV aircraft have this AFT of the NP whereas models should have this in front of the NP or Neutral Point. (or Center of Pressure CP) It should be noted that CCV aircraft have a tailplane significantly SMALLER than would be considered normal. This should be taken into account when designing a MODEL otherwise it may not be large enough for normal control. The WWII Spitfire has a very small tailplane to reduce drag and increase manoevreability.
Terminology There is a lot of confusing terminology regarding this subject and an attempt has been had to clarify these in the notes at the end. Please read these first. In most engineering calculations weights & pressures are assumed to act a one point ie. at the C of G. We know this is not exactly the case as the weight of any object is spread over its entire volume/area, not always evenly, but spread nevertheless. Concentrating this mass or weight at one (imaginary) point makes meaningful calculations possible.
Biplanes and other multi-wing aircraft. I came across these diagrams in an old copy of the Aviation Handbook - Johnson - 1931 and it does seem to be a very authoritative method of determining the location of the MAC in the Z axis for a Biplane. Johnson further states in his book that the Upper Wing carries approx. 58% of the load and the Lower Wing approx. 41%. A very small amount comes from the Fuselage, Struts etc. The C of G should be located at approx. 30% of the mean chord. Optimum range 28% - 33%. The Fig D-1 allows you to compute the location plane of the MAC (in the Z axis) between the Upper and Lower Wing and Fig D-2 provides a solution for value K in D1. Fig A is a repeat of the graphical method of determining the MAC. Fig B is the same but for more complicated wing planforms and Fig C shows the limits for the C of G on various wing positions. Note that you will still need to compute the location of the C of G in the X & Y axis using other sections of this page. D-1 only shows 3 Gap/Chord ratio's. Extrapolate for others.
This article is reproduced with the kind permission of Alasdair Sutherland and should enable any modeller to determine the MAC on any Biplane
Determining the C of G by the use of a Model By building an accurate small scale model of your new model it is possible to determine very accurately where the C of G lies. Make a model (say 1/4) the size of your model as accurately as possible c/w with profile fuselage, tailplane and fin. Make the wings and tail in sheet balsa and round the LE/TE. Test glide the scale model until a nice flat glide results. Remove nose weight until a very slight stall results. This is the AFT position. Now add weight until the glide is unacceptable. This is the most FORWARD position. Check the location on the test model and set the C of G on your full size model somewhere in between these two extremes. At least it should fly with reasonable safety and is better than nothing for very unorthodox or hard to compute models, such as Rear Swept Biplanes or Canards.
Flying Wings & Deltas Use the calculators given below but put in very low values (0.01 for example) for the (non existant) Tailplane. Some of the calculators will not accept zero as a value. The results are confirmed by the positions on models that fly well. Zagi's, Rapier, Delta 363 etc.
Gordon Whitehead - Winning Formula This article appeared in the May issue of Radio Modeller 1994 and is a very comprehensive look at Center of Gravity calculations. Interestingly it covers a wide variety of Multi Wing Aircraft as mentioned above. It probably gives the best method of calculating Centres of Gravity you are likely to find with an accuracy way above anything modellers are likely to require.
Rene Jassien Looking through some old Model Aircraft Magazines (circa 1983) I came across an article by Rene Jassien a well known and very successful competition flyer in the 1980's This article gives a multi-factor method of calculating the Center of Gravity to very precise limits. Specifically designed for competition use where the C of G may be set back as much as 75% from the leading edge of the wings mean chord, or may be even BEHIND the trailing edge. This is due mainly by the use of lifting tailplanes set a long way back from the wings trailing edge. As you can see the theoretical values matched very precisely the actual settings on real well designed and successful models.
Surprisingly when the dimensions of a Frontier Basic Trainer were fed in the solution was 24%, much in line with what one would use for such a model. To make the calculation very simple I have produced an Excel spreadsheet, just feed in the numbers and out pops the solution. Please note that this formula and the models it was designed for are 25 years old and model design and technology may have moved on. Presumably the way aircraft fly has not, so the solutions may still have real value.
** As I was unable to contact Rene Jassien this article is reproduced without his permission.
Barnaby Wainfan is not an author/designer who springs to mind but he has produced an excellent book on "Airfoil Selection" and he has also produced some foils of his own, these are hard to find and may be listed as BW types or as Wainfan. The book can be obtained from http://www.aircraftspruce.com navigate to Books/Videos then Books then Design. The Aircraft Spruce Company site is a mine of information for modellers & full size builders/flyers alike. Well worth a visit.
Model Aircraft Aerodynamics by Martin Simons is one of the best books on the subject. Motorbooks International Wisconsin USA ISBN 0-85242-915-0
See also Dr. Martin Hepperles' web site on Aerodynamics for Model Aircraft Go to the Flying Wings Section on the left.
Links to sites with C of G calculators and further information.
http://www.geistware.com/rcmodeling/index.htm Go to Tips & Techniques/Construction Tips/Calculators/Aircraft Balance Calcs.
Notes on Nomenclature.
Center of Gravity or C of G This is a point on an aircraft that when suspended it would balance in a normal flying attitude. It is should always be on the longitudinal center line (nose/tail) It is the only parameter that can be easily adjusted once the model is designed and built, by adding weight to the nose or tail. To determine a models C of G suspend it from any point by a piece of string, say an undercarriage leg. Project the axis of the string onto the fuselage side and repeat for some other point, say a Wing Tip. Where the projections cross is exactly the C of G
Neutral Point (NP) or Center of Pressure (CP) This is a point on the aircraft's center line where all the lift forces (Wings,Tailplane, Fuselage etc.) are assumed to act. The center of gravity must always be in front of this point by a factor known as the Static Margin. See below. The closer the C of G is to the Neutral Point the less stable is the aircraft. The new Eurofighter probably has its C of G aft of the Neutral Point, so whilst it may need a computer to fly, it is very agile. Models of the Eurofighter, with the C of G in front of the Neutral Point fly like any other Delta or Canard.
Model Aircraft (usually highly refined competition models) with very long moment arms can easily have the Neutral Point well AFT of the wings trailing edge, giving rise to a surprising location for the C of G which may well lie on or even AFT of the wings trailing edge.
The neutral point is quite difficult to calculate but the method in this article gives an excellent practical method. It works by making a card model where the C of G of the wing and tail has been (artificially) re-located to the 25% position, which is now the center of lift or Neutral Point
(Aircrafts) Center of Lift Same as the Neutral Point or Center of Pressure
Static Margin The distance or amount, usually expressed as a %, expressing the location of the Center of Gravity in front (or behind) of the Neutral Point. Usually between 5% and 20%. 5% would be suitable for a highly responsive aerobatic model and 20% for a docile trainer. 15% is a good starting guess. Modern "Fly by Wire" combat aircraft are designed to be unstable as this gives fast control response. The speed of control input required to fly is beyond the capacity of the human brain so a computer actually flies the aircraft interpreting the inputs from the pilot. The simple Dive Test will show if the C of G is about correct. From a good height put the model into a power off vertical dive, with controls at neutral. If the model pulls out by itself the C of G is not far out. If it continues down, pull out quickly and add weight to the nose. Works every time. The Static Margin is expressed as a % of the Wing Chord at the MAC point. So if the wing has a Chord of 10" at the MAC then a Static Margin of 20% will be 2" in front of the C of G.
Wing Chord The distance across a wing parallel to the fuselage center line. Usually identical to the length of a normal wing rib.
Mean Chord The average of all chords.
Chord Line A line joining the leading edge and the trailing edge of an airfoil section. Note as a rule of thumb, most aerofoils only start to produce lift when the Chord line is at 2 degs positive.
Tip Chord The chord at the outer end of the wing.
Root Chord The chord at the inner end of the wing.
Mean Aerodynamic Chord (MAC) A point out from the wings root where all lift (for that 1/2 surface only) is assumed to be located. Once the MAC is located the Aerodynamic Center is located on this line 25% back from the leading edge.
Wing Aerodynamic Center (WAC) For any conventional aerofoil, each wing, tailplane or fin has its center of lift at approx. the 1/4 chord position. i.e. 25% in from the leading edge and at some point out from the wing root. (MAC) With a correctly designed aerofoil this 25% point does not move significantly in normal flight. If it did the aircraft would be very unstable as it would be going in and out of trim all the time as the angle of attack altered. The calculations in the links above determine this point for each lifting surface combining these to give the Neutral Point. This can be determined by the use of nomograms, mathematical calculation or by simple geometric methods. Common examples of the geometric method are shown in the drawings below. Once we have established the Neutral Point and the Static Margin we can determine the Center of Gravity and adjust accordingly.
Tailplane Efficiency See para 2 above. Varies between 30 & 100% dependent on location.
Half Span In all these calculations it is usual to consider one half of the model only. If the total span is 600mm but the fuselage width is 60mm wide at the wing root then the 1/2 span is 600 - 60 / 2 = 270mm. Ignore any Dihedral loss unless significant.
Wing Taper Ratio Tip Chord divided by Root Chord. eg. Root Chord 200mm, Tip Chord 50mm Wing Taper Ratio is 200/50 = 4:1
Aspect Ratio Basically the Span / Chord. eg. A plain rectangular wing of 1000mm Span with a Chord of 50mm has an Aspect Ratio of 1000/50 = 20:1 For tapered, swept wings or delta's you must use the Mean or average Chord value.
Text © Colin Usher 2011 Illustrations © Colin Usher 2011
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical or photocopying, recording or otherwise without the prior permission of the copyright holder. Except for private & non-profit use.