\chapter{Basics of model rocket flight} \label{chap-basics} As rockets and rocket motors come in a huge variety of shapes and sizes, different categories are defined for different levels of rocketry. {\it Model rocketry} itself is governed by the NAR Model Rocket Safety Code~\cite{nar-safety-code} in the U.S. and other similar regulations in other countries. The safety code requires that the model rockets be constructed only of light-weight materials without any metal structural parts, and have a maximum lift-off weight of 1.5~kg. They may only used pre-manufactured motors of classes A--G (see Section~\ref{sec-motor-classes} for the classification). {\it High power rocketry} (HPR) is basically scaled up model rocketry. There are no weight restrictions, and they can use pre-manufactured solid or hybrid rocket motors in the range of H--O. The combined total impulse of all motors may not exceed 81\s920~Ns. {\it Experimental} or {\it amateur rocketry} includes any rocketry activities beyond model and high power rocketry. This may include for example using motor combinations that exceed the limits placed by high power rocketry, building self-made motors or utilizing liquid fueled motors. Finally there is {\it professional rocketry} which is conducted for profit, usually by governments or large corporations. Even though rockets come in many different sizes, the same principles apply to all of them. In this thesis the emphasis will be on model rocketry, but the results are just as valid for larger rockets as long as the assumptions of for example the speed range remain valid. In this chapter the basics of model rocketry and differences to high power rocketry are explained. \section{Model rocket flight} A typical flight of a model rocket can be characterized by the four phases depicted in Figure~\ref{fig-model-flight}: % \begin{enumerate} \item Launch: The model rocket is launched from a vertical launch guide. \item Powered flight: The motor accelerates the rocket during the powered flight period. \item Coasting flight: The rocket coasts freely until approximately at its apogee. \item Recovery: The recovery device opens and the rocket descends slowly to the ground. \end{enumerate} \begin{figure} \centering \epsfig{file=figures/model-flight,scale=0.8} \caption{The basic phases of a typical model rocket flight: 1.~Launch, 2.~Powered flight, 3.~Coasting and 4.~Recovery.} \label{fig-model-flight} \end{figure} Model rockets are launched from a vertical launch guide that keeps the rocket in an upright position until it has sufficient velocity for the fins to aerodynamically stabilize the flight. The NAR safety code forbids launching a model rocket at an angle greater than $30^\circ$ from vertical. A typical launch guide for small rockets is a metal rod about 3-5~mm in diameter, and the launch lug is a short piece of plastic tube glued to the body tube. Especially in larger rockets this may be replaced by two extruding bolts, the ends of which slide along a metal rail. Use of a launch lug can be avoided by a tower launcher, which has 3--4 metal bars around the rocket that hold it in an upright position. After clearing the launch guide, the rocket is in free, powered flight. During this phase the motor accelerates the rocket while it is aerodynamically stabilized to keep its vertical orientation. When the propellant has been used, the rocket is typically at its maximum velocity. It then coasts freely for a short period while the motor produces smoke to help follow the rocket, but provides no additional thrust. Finally, at approximately the point of apogee, a small pyrotechnical ejection charge is fired upwards from the motor which pressurizes the model rocket and opens the recovery device. High-power rocket motors usually have no ejection charges incorporated in them. Instead, the rocket carries a small flight computer that measures the acceleration of the rocket or the outside pressure change to detect the point of apogee and to open the recovery device. Frequently only a small drogue parachute is opened at apogee, and the main parachute is opened at some pre-defined lower altitude around 100--300 meters. The typical recovery device of a model rocket is either a parachute or a {\it streamer}. The parachutes are usually a simple planar circle of plastic or fabric with 4--10 shroud lines attached. A streamer is a strip of plastic or fabric connected to the rocket, intended to flutter in the air and thus slow down the descent of the rocket. Especially small rockets often use streamers as their recovery device, since even light wind can cause a light-weight rocket with a parachute to drift a significant distance. \section{Rocket motor classification} \label{sec-motors} \label{sec-motor-classes} The motors used in model and high power rocketry are categorized based on their total impulse. A class `A' motor may have a total impulse in the range of 1.26--2.50~Ns. Every consecutive class doubles the allowed total impulse of the motor. Thus, a B-motor can have an impulse in the range 2.51--5.00~Ns and a C-motor in the range 5.01--10.0~Ns. There are also classes \half A and \quarter A which have impulse ranges half and one quarter of those of an A-motor, respectively. Commercial rocket motors are available up to class~O with a total impulse of 30\s000~Ns~\cite{all-certified-motors}. Table~\ref{tab-motor-classes} lists the impulse ranges for model and high-power rocket motors. \begin{table} \caption{Total impulse ranges for motor classes \quarter A--O.} \label{tab-motor-classes} \begin{center} \begin{tabular}{cr@{--}l|cr@{--}l|cr@{--}l} \hline \quarter A & 0.0 & 0.625~Ns & E & 20.01 & 40.0~Ns & K & 1280.01 & 2560~Ns \\ \half A & 0.626 & 1.25~Ns & F & 40.01 & 80.0~Ns & L & 2560.01 & 5120~Ns \\ A & 1.26 & 2.50~Ns & G & 80.01 & 160~Ns & M & 5120.01 & 10240~Ns \\ B & 2.51 & 5.00~Ns & H & 160.01 & 320~Ns & N & 10240.01 & 20480~Ns \\ C & 5.01 & 10.0~Ns & I & 320.01 & 640~Ns & O & 20480.01 & 40960~Ns \\ D & 10.01 & 20.0~Ns & J & 640.01 & 1280~Ns & \\ \hline \end{tabular} \end{center} \end{table} Another important parameter of a rocket motor is the thrust given by the motor. This defines the mass that may be lifted by the motor and the acceleration achieved. Small model rocket motors typically have an average thrust of about 3--10~N, while high-power rocket motors can have thrusts in excess of 5\s000~N. The third parameter used to classify a model rocket motor is the length of the delay between the motor burnout and the ignition of the ejection charge. Since the maximum velocity of different rockets using the same type of motor can be vastly different, also the length of the coasting phase varies. Therefore motors with otherwise the same specifications are often manufactured with several different delay lengths. These delay lengths do not apply to high-power rocket motors, since they do not have ejections charges incorporated in them. Model rocket motors are given a classification code based on these three parameters, for example ``D7-3''. The letter specifies the total impulse range of the motor, while the first number specifies the average thrust in Newtons and the second number the delay of the ejection charge in seconds. The delay number can also be replaced by `P', which stands for {\it plugged}, \ie the motor does not have an ejection charge. Some manufacturers may also use an additional letter at the end of the classification code specifying the propellant type used in the motor. Even motors with the same classification code may have slight variations to them. First, the classification only specifies the impulse range of the motor, not the exact impulse. In principle, a D-motor in the lower end of the range might have a total impulse only 1~Ns larger than a C-motor in the upper end of its range. Second, the code only specifies the average thrust of the motor. The thrust rarely is constant, but varies with time. Figure~\ref{fig-thrust-curve} shows the typical thrust curve of a small black powder rocket motor. The motors typically have a short thrust peak at ignition that gives the rocket an initial acceleration boost before stabilizing to a thrust level a little below the average thrust. Statically measured thrust curves of most commercial rocket motors are readily available on the Internet~\cite{thrust-curve-database}. \begin{figure} \centering \epsfig{file=figures/motors/D12-thrustcurve,width=9cm} \caption{A typical thrust curve of an Estes D12-3 rocket motor and its average thrust.~\cite{D12-curve}} \label{fig-thrust-curve} \end{figure} Also the propellant type may affect the characteristics of the motor. Most model rocket motors are made up of a solid, pyrotechnical propellant---typically black powder---that is cast into a suitable shape and ignited on launch. Since the propellant burns on its surface, different thrust curves can be achieved by different mold shapes. % vesiraketit! A significantly different motor type, {\it hybrid motors}, were commercially introduced in 1995. These motors typically include the propellant and oxidizer in different states, typically a composite plastic as the fuel and a separate tank of liquid nitrous oxide ($\rm N_2O$) as the oxidizer. The plastic on its own does not burn very well, but provides ample thrust when the nitrous oxide is fed through its core. The nitrous oxide tank is self-pressurized by its natural vapor pressure. However, since temperature greatly affects the vapor pressure of nitrous oxide, the thrust of a hybrid motor is also diminished if the oxidizer is cold. On the other hand, the motor will burn longer in this case, and since nitrous oxide is denser when cold, the motor may even yield a greater total impulse. The significance of this effect was observed when analyzing the video footage of the launch of the first Finnish hybrid rocket, ``Haisunäätä''~\cite{haisunaata-launch}. The average thrust during the first 0.5~seconds was determined to be only about 70~N, whereas the static tests suggest the thrust should have been over 200~N. Instead, the motor burned for over 10~seconds, while the normal thrust curves indicate a burning time of 5--6~seconds. This shows that the temperature of the hybrid motor oxidizer can have a dramatic effect on the thrust given by the motor, and the static test curve should be assumed to be valid only in similar operating conditions as during the test. One further non-pyrotechnical rocket type is {\it water rockets}. These are especially popular first rockets, as they require no special permits and are easy to construct. The water rocket includes a bottle or other chamber that has water and pressurized air inside it. On launch the pressure forces the water out of a nozzle, providing thrust to the rocket. While simulating water rockets is beyond the scope of this thesis, it is the aim that methods for modeling water rockets can be added to the produced software in the future. \section{Clustering and staging} Two common methods used to achieve greater altitudes with model rockets are {\it clustering} and {\it staging}. A cluster has two or more rocket motors burning concurrently, while staging uses motors that burn consecutively. The motor configuration of a cluster and staged rocket is depicted in Figure~\ref{fig-cluster-stages}. When a cluster is launched, the total thrust is the sum of the thrust curves of the separate motors. This allows greater acceleration and a greater liftoff weight. Staging is usually performed by using zero-delay motors, that ignite the ejection charge immediately at burnout. The ejection charge fires towards the upper stage motor and ignites the next motor. High power motors with no ejection charges can be clustered by using an onboard accelerometer or timer that ignites the subsequent stages. Staging provides a longer duration of powered flight, thus increasing the altitude. \begin{figure} \centering \parbox{65mm}{\centering \epsfig{file=figures/motors/cluster,width=60mm} \\ (a)} \hspace{10mm} \parbox{40mm}{\centering \epsfig{file=figures/motors/staged,width=30mm} \\ (b)} \caption{The motor configuration for (a) a cluster rocket and (b) a two-staged rocket.} \label{fig-cluster-stages} \end{figure} Clustering provides a greater acceleration at launch, but staging typically provides greater altitude than a cluster with similar motors. This is because a clustered rocket accelerates quickly to a greater speed thus also increasing the aerodynamic drag. A staged rocket has a smaller thrust for a longer period of time, which reduces the overall effect of drag during the flight. \section{Stability of a rocket} \label{sec-stability} When designing a new rocket, its stability is of paramount importance. A small gust of wind or some other disturbance may cause the rocket to tilt slightly from its current orientation. When this occurs, the rocket centerline is no longer parallel to the velocity of the rocket. This condition is called flying at an {\it angle of attack $\alpha$}, where $\alpha$ is the angle between the rocket centerline and the velocity vector. When a stable rocket flies at an angle of attack, its fins produce a moment to correct the rocket's flight. The corrective moment is produced by the aerodynamic forces perpendicular to the axis of the rocket. Each component of the rocket can be seen as producing a separate normal force component originating from the component's CP, as depicted in Figure~\ref{fig-normal-forces}. \begin{figure} \centering \epsfig{file=figures/aerodynamics/component-normal-forces,width=130mm} \caption{Normal forces produced by the rocket components.} \label{fig-normal-forces} \end{figure} The effect of the separate normal forces can be combined into a single force, the magnitude of which is the sum of the separate forces and which effects the same moment as the separate forces. The point on which the total force acts is defined as the center of pressure or the rocket. As can be seen from Figure~\ref{fig-normal-forces}, the moment produced attempts to correct the rocket's flight only if the CP is located aft of the CG. If this condition holds, the rocket is said to be {\it statically stable}. A statically stable rocket always produces a corrective moment when flying at a small angle of attack. The argument for static stability above may fail in two conditions: First, the normal forces might cancel each other out exactly, in which case a moment would be produced but with zero total force. Second, the normal force at the CP might be in the wrong direction (downward in the figure), yielding an uncorrective moment. However, we shall see that the only component to produce a downward force is a boattail, and the force is equivalent to the corresponding broadening of the body. Therefore the total force acting on the rocket cannot be zero nor in a direction to produce an uncorrective moment when aft of the CG. The {\it stability margin} of a rocket is defined as the distance between the CP and CG, measured in {\it calibers}, where one caliber is the maximum body diameter of the rocket. A rule of thumb among model rocketeers is that the CP should be approximately 1--2 calibers aft of the CG. However, the CP of a rocket typically moves upwards as the angle of attack increases. In some cases, a 1--2 caliber stability margin may totally disappear at an angle of attack of only a few degrees. As side wind is the primary cause of angles of attack, this effect is called {\it wind caused instability}~\cite{galejs}. Another stability issue concerning rocketeers is the {\it dynamic stability} of a rocket. A rocket that is statically stable may still be poor at returning the rocket to the original orientation quickly enough. Model rockets may encounter several types of dynamic instability depending on their shape, size and mass~\cite[pp.~140--141]{stine}: % \begin{enumerate} \item {\it Too little oscillation damping.} In short, light-weight rockets the corrective moment may significantly over-correct the perturbation, requiring a corrective moment in the opposite direction. This may lead to continuous oscillation during the flight. \item {\it Too small corrective moment.} This is the case of over-damped oscillation, where the corrective moment is too small compared to the moment of inertia of the rocket. Before the rocket has been able to correct its orientation, the thrust of the motors may have already significantly affected the direction of flight. \item {\it Roll-pitch coupling.} If the model has a natural roll frequency (caused \eg by canting the fins) close to the oscillation frequency of the rocket, roll-pitch resonance may occur and cause the model to go unstable. \end{enumerate} By definition, dynamic stability issues are such that they occur over time during the flight of the rocket. A full flight simulation that takes into account all corrective moments automatically also simulates the possible dynamic stability problems. Therefore the dynamic stability of rockets will not be further considered in this thesis. For an analytical consideration of the problem, refer to \cite{advanced-model-rocketry}.