Richard Nakka’s Experimental Rocketry Web Site

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Introduction to Rocket Design

Appendix D

Estimating Centre of Gravity of a Rocket

Introduction

The location of the Centre of Gravity (C.G.) is the point of a body around which the resultant torque due to gravity vanishes. In other words, as applied to a rocket, the balance point. If the rocket is hung from a string, as shown in Figure 1, it will be balanced, level and stationary if attached at the C.G. location.

Although centre of gravity has traditionally been the term used in model, HPR and EX rocketry, the technically preferred term is Centre of Mass, as it is more fundamental (applies even when gravity is not uniform). However, we’ll stick to the traditional term of Centre of Gravity, or its abbreviation, C.G.

Figure 1: Determining balance point of model rocket
(ref. https://waterbottlerockets.weebly.com/design---cg--cp.html)

Using a string to find the balance point works well for model rockets, but this technique is less practical for larger and heavier HiPower and EX rockets. For my EX rockets, I simply balance the rocket on a knife edge. Not an actual knife, but on a wedge (such as a piece of angle aluminum).

Measuring the C.G. location of the fully-assembled rocket is important for verification the static stability margin. However, during the design phase of a rocket, measurement is not a viable option. Predicting the location of the C.G. during the design process helps ensure a good stable design with a stability margin within the range of the design goal.

Method

Centre of gravity of a rocket is determined by calculating a weighted average of the positions of all the parts that make up the rocket. Basically, you sum up the product of each part’s mass and its position, then divide by the total mass of the parts. The position of any particular part is with respect to some chosen reference point, or datum. For a rocket, the datum is typically the tip of the nosecone with positive x aftward along the rocket’s longitudinal (roll) axis. This is known as nose-to-tail axis system. However, the datum can be any convenient location, such as the aft end of the body with positive x forward. This is tail-to-nose axis system. These two possible coordinate systems are shown in Figure 2.

Figure 2: Coordinate systems for determing C.G.

Note that y and z axes are also shown, whereby the z-axis is normal to the x-y plane. In the nose-to-tail system, the z-axis is pointing toward the observer and with the tail-to-nose system, the z-axis is pointing away from the observer.

Equation 1

In equation form, specifying a coordinate system with x along the rocket’s longitudinal (roll) axis, such as shown in Figure 2, the x-location of the composite C.G. is given by:

Or in summation notation:

where m_i is the mass of a particular part, and of which there are n parts in total, and x_iis the distance along the x-axis of the centre of gravity of that particular part.

The term composite C.G. implies that the body is comprised of many items of mass, which is, of course, applicable to any rocket. For convenience, we’ll simply use the term C.G. (or C.G. of the rocket).

This same method can also be used to calculate y_CG and z_CG. These indicate the location of the C.G. with respect to the roll axis of the rocket. This is applicable to rocket dynamic stability.

Consider the design of the simplified rocket illustrated in Figure 3, with the datum being the tip of the nosecone:

Figure 3: Example rocket

The C.G. location for this example is given by:

The estimated x-location of the parts individual C.G.s are presented in Table 1, together with their estimated masses and calculation of the predicted C.G. of the rocket being designed (Figure 4).

Table 1: Example rocket C.G. calculation

Figure 4: Example rocket showing calculated C.G. location

Equation 1 tells us something important, although it is not obvious when an arbitrary datum point such as nosecone tip is chosen. The location of the composite C.G. is comprised of the sum of the product of each part’s mass and x-position, which we will refer to as the inertial torque:

T_i = m_i ´ x_i

Both mass of the part and its distance from the C.G. are seen to be equally important. The effect of this is apparent if instead we use the C.G. location as our datum. Of course, we do not know ahead of time where our C.G. is located, this exercise is simply to illustrate. As such, we end up with that presented in Table 2:

Table 2: Example rocket using known C.G. location as our datum

We are solely interested in the magnitude, or absolute value, of the inertial torque values. As such we can drop the negative sign of any values, which is merely a consequence of our chosen datum location. The farther a part is located from the composite C.G., the more influence its contribution of its gravitational or inertial torque about the C.G. This is shown in Table 3 which has the parts sorted in order of influence.

Table 3: Example rocket with inertial torque sorted

For this example rocket, the motor plus propellant has the most profound effect on the C.G. location. The body, on the other hand, even though its mass is greater, has only one half the influence. Two other things to note. The payload, whose C.G. is close to the rocket’s C.G., has very little effect. Increasing or decreasing payload mass will not affect the balance of the rocket much. The lightweight transmitter, whose mass is only 2% of the rocket, contributes 6% of the balancing torque, as it is located far from the rocket C.G. The takeaway is that during the design phase, both the mass of a part and its location need careful consideration, as both figure prominently in the determination of the C.G. of the rocket.

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Estimating the C.G. of individual parts that comprise the rocket can be straightforward for certain parts such as those that have a symmetric geometry, such as the rocket body, or complicated, for parts with a complex shape such as a nosecone or payload. There are two approaches to this problem, both equally valid and important:

· Hand calculation

· CAD modeling

Hand calculations allow the designer to more quickly assess a design, especially when done in conjunction with a spreadsheet app such as MS Excel. CAD modeling provides accurate mass and C.G. information once a design becomes more refined.

Hand calculation method is discussed first.

Tubes and solid cylinders

For a tube of uniform thickness, such as a body tube, or a solid cylinder such as a propellant grain, the C.G. position is simply the geometrical centre. Mass is calculated as volume times density (r). For a tube or for a solid cylinder (D_i = 0):

m = ¼ r (D² – Di²) L

Density is in units of mass per unit volume, such as lbm/in³ or kg/m³.

Nosecone

The nosecone for a rocket can be selected from a vast variety of shapes. A nosecone can be solid, which is typical for a model rocket, or hollow, which is usually the case for HiPower or EX rockets. As such, there is no single approach to determining the mass and C.G. of a nosecone. For non-standard nosecone shapes, that follow a known mathematical function, calculus or numerical integration can be used, especially for those that are hollow.

For a solid straight conical nosecone, such as that shown as figure A of Figure 5, the volume is given by:

V = 1/12 π D² H

Figure 5: Conical nosecone shapes

The C.G. (also referred to as centroid of a volume) of a solid cone is located at a distance ¼ H from the base.

A hollow conical nosecone, such as that shown as B in Figure 5, the net volume is that of figure A minus the smaller cone of figure C :

V_B = V_A - V_C

as such, for a hollow cone:

Figure 6 shows some of the more commonly used shapes for nosecones other than conical. For a complete description of these shapes, refer to Resource 18 (The Descriptive Geometry of Nose Cones).

Figure 6: Various nosecone shapes

The volume and centroid of solid shapes are given below, where D = base diameter and H = height. The location of the centroid (x_CG) is with respect to the base of the nosecone.

Solid Parabolic cone

Solid Paraboloid

Solid Ellipsoid

Solid Tangent Ogive

where f = R/D and . The dimension R is the radius of the ogive, as shown in this figure.

Solid Power Series

where n = power exponent i.e.

n = 0.5 for ½ power (paraboloid) shape

n = 0.75 for ¾ power shape

n = 1.0 for a straight conical shape

Solid Secant Ogive

Secant ogive is essentially a tangent ogive with a truncated base such that the base is not tangent to the rocket body. Resource D1 provides a method to calculate volume and centroid.

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The above formulas are for solid shapes. If a nosecone is hollow, the same method for calculating volume can be applied as for a hollow straight conical nosecone, as given earlier:
V_B = V_A - V_C

Figure 7 illustrates a nosecone constructed of two identical volumetric shapes except the interior shape is smaller, whose volume is subtracted from the larger volume.

Figure 7: Hollow parabolic cone (section cut view)

To determine the location of the C.G. of a hollow nosecone, the following factor is applied to the C.G. location for solid shapes:

where s is the scale factor as illustrated in Figure 8.

Figure 8: Scale factor for hollow nosecones

As such, for a solid cone (s = 0) the term in brackets is equal to unity and for a thin shell (s » 1), the term approaches 4/3.

As stated earlier, mass is calculated as volume times density (r)

Fins

For two-dimensional parts with a constant thickness, such as fins, the C.G (in the plane of the part) corresponds to the centroid of the shape. As an example, consider the trapezoid shaped fin (A-B-C-D) in Figure 9 below, defined by dimensions a, b, c and h:

Figure 9: Centroid of a trapezoid shape

Note that this is also valid for swept-back fins, whereby dimension c is a negative number.

The C.G. corresponds to the centroid if the fin has constant thickness.

Other Items of Mass

As discussed earlier, if an item of mass, such as AvBay, lies close to the (expected) C.G. of the rocket, its influence on the overall C.G. is slight. As such, simply using the geometric centre of the item as its C.G. location is generally sufficient. On the other hand, items located further from the C.G. of the rocket require more care in estimating the item’s C.G. location. If the item is comprised of smaller parts, Equation 1 can be used to accurately derive the C.G. of the assembled parts.

CAD Modeling

CAD modeling has the capability of providing highly accurate estimations of the mass of a rocket’s components as well as their C.G. position. Density values can be assigned to materials which are utilized to compute mass of the various parts. The C.G. of the complete rocket assembly can also be readily computed. Hardware items such as fasteners and eyebolts are available as STEP files from vendors such as McMaster-Carr. This helps facilitate modeling and allows for good fidelity of the model.

CAD modeling proves it worth as a useful design tool as a rocket design becomes more refined and provides a blueprint for constructing the rocket assembly. Figure 10 shows my Arrow rocket that was modeled in CAD after performing basic hand calculations to size the rocket. Figure 11 shows the contents of the AvBay. Each part is assigned a material, which allows for each part to have a representative mass. For parts such as a battery or circuit component, effective density can be used, whereby effective density is the measured weight of the part divided by its modeled volume. Lightweight parts such as parachute and tethers are not modeled, per se, but can be modeled as a simple lumped mass at its estimated C.G. location. Figure 12 shows the CAD computed C.G. of the complete Arrow rocket.