Nomenclature: P = Momentum m = Mass u = Velocity F = Force t = Time Cd = Drag Coefficient Cl = Lift Coefficient
The flight dynamics of a rocket or most flight vehicles can be modeled using the conservation laws of nature.
Conservation of Momentum: Momentum is defined as the quantity of motion of a moving body or system. It is measured as the product of the body's mass and velocity. P = m x u
A force is either a push or pull exerted on a body or system. Isaac Newton discovered that the forces acting on a body is equal to the change in momentum of the body over time. F = dP/dt
Applying the above equation one can model the flight dynamics of a rocket. The external forces commonly acting on flight vehicles in the atmosphere are thrust, aerodynamic forces, weight. Other forces such as wind or electromagnetic forces can also be applied, but at times they tend to be small and negligible.
Thrust is the force produced by the power plant of the vehicle such as a rocket engine. It tends to act in the direction inward normal to the nozzle exit area. The thrust force in chemical rocket engine is developed by converting the chemical energy of a source into thermal energy through combustion. The combustion gases are expanded through a nozzle which converts the thermal energy into kinetic energy, developing thrust as the gases exit the nozzle. In terms of momentum, the fast moving gases produced imparts momentum onto the engine as it exits the nozzle, producing thrust that propels a vehicle forward. The thrust of rocket engine can be expressed as: T = mdot*ve+ (Pe-Po)*Ae where mdot is the rate at which mass is leaving the rocket, ve is the velocity at which the mass is exiting the rocket, Pe is the static pressure of mass at the exit of the nozzle, Po is the back pressure impart by the atmosphere, and Ae is the exit area of the nozzle. The mass leaving the rocket is the mass of the hot gases exiting at high velocities from the nozzle. The rate at which mass is leaving the rocket or the mass flowrate is usually constant is rockets if transient events are small and negligible. The expression for thrust can be simplified into the following: T =mdot*c where c is the effective exhaust velocity, and it combines the pressure terms and exit velocity. When Pe is equal to Po, c is equal to ve. This known as perfect expansion, and it is when the thrust is maximum.
The two aerodynamic forces acting on a vehicle during flight are the drag force and lift force.
D = 0.5*Cd*rho*u^2*A
L = 0.5*Cl*rho*u^2*A
Since the vehicle has mass and it is in a gravitational field imparted by the earth, the weight of vehicle is another external force acting on the vehicle. The weight of a body near a gravitational field can be expressed as the mass of the body times the acceleration due to the field. The acceleration due to earth's gravitational field is 9.81 m/sec^2 at the surface. The weight force acts in the direction of the acceleration vector.
W = m*g
The free body diagram of a rocket in atmospheric flight can then be expressed as follow when applying the above external forces:
Free Body Diagram
Applying the conservation of momentum to the free body diagram, two sets of differential equations in the normal and tangential direction to the rocket flight path are derived.
These equations can be used to analyze the kinetic and kinematic behavior of the rocket in flight, and predict its flight path.
Software and Simulation approach
Software exist that apply the laws of conservation of momentum and energy, and numerically solve the differential equations of rocket flight to predict rocket kinematic and kinetic behavior. RasAero II, OpenRocket, and RockSim were the three software of choice used to model the hybrid engine rocket. The simulations were used to determine engine design specifications such as thrust to meet desired goals. The desired goals are that the engine must be able to deliver a 140 pound rocket to a height of 30,000 ft exactly under a total impulse of 9,208 lb-sec, while having a lift off velocity of at least 100 ft/sec for rocket stability. Multiple simulations were performed to design and validate engine specifications to meet desired goals.
The procedure taken for the thrust, kinematic, and kinetic analysis of the rocket is shown below.
Regression rate analysis- direct to regression rate analysis of engine
Thrust curve development- use results from the regression rate analysis, and combine with tctracer to develop a thrust curve to be used in the rocket simulation softwares.
Rocket model development using RasAero, RockSim, and OpenRocket
Simulation and analysis of thrust curve rocket model