IOWA STATE UNIVERSITY - Aerospace Engineering Notes

AER E 1600: Lecture 7 Overview

• Presenter: Carolyn Riedel
• Focus: Programming V - A Beginner's Guide to Rocket Math
• Lecture Content:
  • Thrust Specific Fuel Consumption (TSFC)
  • Change in mass of the rocket
  • Force Due to Gravity
  • Forces on the Rocket
  • Acceleration of the rocket
  • Velocity of the rocket
  • Position of the rocket
  • Demonstration of the Programming Project

Thrust Specific Fuel Consumption (TSFC)

• Definition: TSFC is a measure used to indicate rocket efficiency, specified as the rate of fuel consumption per unit thrust.
• Key Points:
  • Important parameters in rocket propulsion include thrust and fuel efficiency.
  • TSFC is critical for understanding the efficiency of rocket engines.
  • Units: Common units for TSFC include kilograms per Newton per second in the Metric system or pounds per pound-force per hour in English units.

Change in Rocket Mass Over Time

• Concept: As a rocket burns fuel, its total mass decreases.
• Mathematical Representation: The change in mass over time can be expressed using the equation: $\frac{dm}{dt} = -TSFC * T$
  • Here,
    • TSFC is the thrust specific fuel consumption,
    • T is the thrust of the rocket.
• Using Euler's Method: The change in mass can be reformulated as: $\delta m = (-TSFC * T)\delta t$
  • Mass of fuel at any instant in time is given by:
    $m<em>{fuel}(t) = m</em>{fuel}(t_0) + \delta m$
  • Total mass at any point:
    $m<em>{rocket}(t) = m</em>{initial} + m_{fuel}(t)$

Rocket Thrust and Thrust Direction

• Understanding Thrust: Thrust is a vector and has both a magnitude (Force) and direction.
• Equation for Thrust Direction: $T_{direction} = \begin{pmatrix} cos(\theta) \ sin(\theta) \end{pmatrix}$
  • Direction is defined in terms of an angle ( \theta ) with reference to axes.

Rocket Global Position and Gravity

• Gravity Force Calculation:
  • Gravitational force can be determined with:
    $F<em>{gravity} = \frac{G m</em>1 m_2}{r^2}$
  • Here,
    • G is the gravitational constant,
    • m1 and m2 are the masses,
    • r is the distance between their centers.
• Direction of Gravity: Represented as:
  $G_{direction} = - \frac{position\, vector}{|position\, vector|}$

Forces on the Rocket

• Total Forces Acting: The net force on the rocket is given by the combination of thrust and gravitational force:
  $F<em>{net} = T</em>{direction} + G_{direction}$
• Newton’s Second Law: This relationship can be analyzed using:
  $\Sigma F = m a$

Acceleration Over Time

• Calculation of Acceleration:
  • Referring to previous equations, acceleration can be expressed as:
    $a = \frac{\Sigma F<em>{net}}{m</em>{rocket}}$

Velocity Over Time

• Velocity Relation to Acceleration: Velocity can be derived using the relationship from physics:
  $a = \frac{dv}{dt}$
• In Terms of Forces: Therefore, we have:
  $a = \frac{\Sigma F<em>{net}}{m</em>{rocket}}$

Velocity Vector Change

• Using Euler’s Method: The change in velocity vector is expressed as:
  $\delta v = \frac{\Sigma F<em>{net}}{m</em>{rocket}} \delta t$
• Velocity Update: The new velocity vector at each timestep can therefore be expressed as:
  $v<em>{new} = v</em>{current} + \delta v$

Position Vector Update

• Position Relation to Velocity: Position is related to the velocity by the equation:
  $vel = \frac{dx}{dt}$
• Updating Position with Euler’s Method: Changes to the position vector can be represented as: $\delta position = vel \delta t$
  • Continuous position update would be:
    $position<em>{new} = position</em>{current} + \delta position$

Programming Project Demonstration

• A programming project demo will provide an example of the expected outcomes based on the discussed principles and equations.

References

• Slides include images and credits (e.g., NASA's Saturn V rocket carrying humans to the moon).
• Referenced materials from NASA/Kim Shiflett for relevant visual aids and further contexts.