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Numerical determination of the center of gravity in rotorcraft |
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Por Antonio Angulo Álvarez The September and October 1997 issues of Rotorcraft Magazine publish several methods for establishing the center of gravity in a rotorcraft. The attached notes provide an additional method, mainly numerical, for determining the location of said center of gravity. DATA Prior to starting the calculation, the following data must be available:
OPERATIONS With the three rotorcraft wheels (two main wheels and one front wheel) on the same level, measure the force applied by the front wheel resting against the ground. This can be easily done with the use of a bathroom weighing scale. This force is P, and must be measured in the same units used for defining the weight W. Kilograms are used in the example. The Q force (similar to the P force) applied on the ground (or on the weight) by the front wheel must also be established when the wheel is positioned at a height h, which is below the height of the back wheels, as can be seen in Figure 2.
DEFINITIONS With the above data, establish the center of gravity (CG) based on the distance between the vertical of said center and the vertical of the main wheel center O. This distance is identified by x. The height of said center of gravity over the horizontal of the main wheel O center is identified by z, as shown in Figure 1. The horizontal at point O, center of rear wheels, is slanted when performing the second operation. Angle a (alpha) defines this slant, also shown in Figure 2. CALCULATIONS Based on the above quantities, magnitude x is obtained applying following formula:
This is a well-known procedure, and therefore no further information need be provided. To obtain length z, apply the distances shown in Figure 3. Using a statics basis, establish this Formula: shown in the graph in Figure 3: Fig. 3
In the formula, replace value x with the value derived from mentioned Formula to reach an expression defining z, according to :
This formula includes the W, P, Q, a and d data, in addition to the angle a trigonometric tangent. Some difficulty is involved in calculating angle a, as a result of the main wheels and the front wheel having different radii. However, it is a fact that most present-day rotorcraft are fitted with wheels of different sizes, so therefore it is essential to perform the calculations bearing in mind said differences in radii. If the lengths of the two dimensions shown to the right and to the left of Figure 3 are equaled:
is obtained wherein angle a is defined by its trigonometric lines (sine and cosine) as a function of the data and height h. Through appropriate following mathematical operations:
the involvement of angle a may be eliminated, thereby achieving an expression for center of gravity height z by applying the last formula included in the expression:
and the problem is thus solved. This last formula, however, is somewhat complicated. The solution may be simplified by using the auxiliary values (m, n and s) indicated as follow:
So that the practical expression of magnitude z is defined through the formula:
Calculating this formula may also seem awkward, although with the aid of a hand calculator, no matter how elemental, it becomes readily available. GRAPH IN FIGURE 4 Fig. 4 The angle a trigonometric tangent may be also educed by using the graph in Figure 4. All that is needed is to search for the vertical reference, namely the value of the fraction h + d divided by a, which defines the height at which the point being searched is located; furthermore, in the horizontal reference, indicate the abscissa d divided by a. The resulting point will be located between the two slanting lines which define specific values of the a tangent. The value of the tangent in question is obtained through interpolation. This value, applied to Formula
provides center of gravity height z. SIMPLIFICATIONS It may be assumed, in approximate terms, that the radii of the three wheels in the rotorcraft are the same. If such is the case, magnitude d would be zero, the used formula
becoming greatly simplified, and resulting the formula :
The maximum simplification is obtained when height h is a small magnitude if compared to length a, namely in the order of 10% of said length. In this case, the formula would be
this being the most simple formula. EXAMPLE An example is analyzed hereunder designed to show the use of the above mentioned formulation, although this does not apply to any actual case. The following values are assumed: W = 310 kg a = 1.70 m R = 0.20 m r = 0.15 m (d = R - r = 0.05 m) P = 20 kg The graphic included in Figure 4. If the wheels are placed at a height h difference of 0.30 m, weight Q increases to 35 kg. COMMENTS The values of x and z have been deduced, and it may be said that the values defined in the two first operations are absolutely exact, since they have been obtained numerically. They derive from applying Formulas:
The following are the resulting values: x = 0.110 m z = 0.4520 m When applying Formula
the solution must be not considered to be exact, since finding the auxiliary value (tang a) requires the use of the graph in Figure 4, and it is a fact that the appreciation of a point in a graph is theoretically exact, although in practice this is not precisely so. In spite of this, the error perceived is small, since the value of z thus obtained is 0.4513 m, differing only 0.0007 m from the exact value - less than 1 mm. The use of Formula
involves ignoring the fact that the diameters of the main wheels and front wheel are different. Save for this, the formula is exact. The result obtained for z is 0.4588 m, differing from the exact value in only 0.0068 m, or almost 7 mm, thus showing that said Formula is applicable in practice, despite its simplicity. On ending, Formula
is merely approximate by definition. However, its application reveals a value of 0.4661 m for z, the resulting error amounting to 0.0141 m, or slightly above a half inch. FINAL CONSIDERATIONS It must be pointed out in relation to the procedure described above that the exact operations for establishing the weights of the front wheel do not seem to be difficult, regardless of whether the rotorcraft is level or whether the main wheels are higher. The numeric calculations can also be performed by using a manual calculator, and it is believed that the overall system as described is truly usable. Madrid, September 2000 |
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