This document summarizes a student project that simulated the longitudinal dynamics of a Boeing 747 aircraft. The objectives were to simulate the short period and phugoid modes, calculate errors between exact and approximate solutions, and visualize the time histories. The key steps included: 1) Developing mathematical models of the aircraft equations of motion and decoupling into longitudinal dynamics. 2) Creating a Simulink model to simulate the short period and phugoid modes. 3) Calculating natural frequencies, damping ratios, and percentage errors between exact and approximate solutions. 4) Visualizing the time histories of parameters like pitch angle and airspeed. The project concluded that short period approximations were accurate while phugoid approximations produced

1. SIMULATION OF
AIRCRAFT DYNAMICS-
LONGITUDINAL MODE
Minor Project

2. Group description
Name Roll No. Batch
Mahendra Gehlot 15/551 A1
Mousam Kumari 15/562 A1
Krishna Shekhawat 15/544 A1
Ishank Srivastava 15/679 A1
Under the supervision of –
Mr. Mohammed Shahid
Assistant Professor
Rajasthan Technical
University, Kota
Submitted to-
Mr. Brijesh Tripathi
Mr. Anshul Khandelwal

3. Objectives
• Simulation of Aircraft Dynamics (Longitudinal Mode) for Boeing 747.
• Error calculation of exact solution to approximate solution for short
and phugoid mode.
• Visualization of Aircraft Dynamic time history of various parameter of
longitudinal mode for exact solution (Boeing 747).

4. Project outline
Aircraft
Equations of
motion
Decoupling of
EOM
Longitudinal
EOM
Selection of
aircraft
Step
1 Mathematical
Modelling on
MATLAB
Short Period
Approximation
Phugoid
Period
Approximation
Step
2 Simulink
Model
Results
Error
Calculation
Step
3 Visualisation
of Simulation
Conclusion
Step
4

5. Introduction
Aircraft Stability
Static Stability Dynamic Stability
Aircraft stability
- Ability of aircraft to correct for conditions that are acting
on it
Initial tendency of an aircraft to
return to its original position when
it’s disturbed
How airplane responds over time
to disturbance

6. Various Modes of Dynamics Stability
Longitudinal mode
- Phugoid (longer period)
oscillations
- Short period oscillations
Lateral-Directional Modes
- Roll subsidence mode
- Dutch roll mode
- Spiral divergence

7. Aircraft Equations of Motion
There are six equation of motion of aircraft consist of:
- Longitudinal Equation of Motion
- Lateral and Direction Equation of Motion
• The six aircraft equation of motion (EOM) can be decoupled into two
sets of three equations.
• These are the three longitudinal EOM and the three lateral-directional
EOM.
• This is convenient in that it requires only three equations to be solved
simultaneously for many flight conditions.

8. Longitudinal Equations of Motion
One way of the thinking of the longitudinal EOM is to picture an aircraft with its xz
plane coincident with an xz plane fixed in space.
Longitudinal motion consists of those movements where aircraft would only move
within that xz plane, that is, translation in the x direction, translation in the z
direction and rotation about y axis.

9. Longitudinal Equations of Motion
The three longitudinal EOM consist of the x force, and z force, and y moment
equations
Where U,V, and W are the velocities in the x, y, and z body axes, respectively.
P,Q, and R are the roll, pitch and yaw rates respectively.

10. Simulation Model : Boeing 747-200

11. Approximations
Short Period Approximation
 The period is so short that the speed
does not have time to change, so the
oscillation is essentially an angle-of-
attack variation.
 The motion is a rapid pitching of the
aircraft about the centre of gravity.
 Its damping ratio and natural frequency
is high.
 Percentage error from exact Solution is
negligible.
Phugoid Approximation
 Large-amplitude variation of air-speed,
pitch angle, and altitude, but almost no
angle-of-attack variation.
 It has a nearly constant angle of
attack but varying pitch, caused by a
repeated exchange of
airspeed and altitude.
 Its damping ratio and natural frequency
is low.
 Percentage error is approximately 30-
40% compared with exact solution.

12. Simulink Model (linear) – extended from MATLAB workspace

13. Exact Solution

14. Exact Solution

15. Results
 Exact Solution
Mode
Natural
Frequency
Damping
Ratio
Short Period 1.3201 0.3555
Phugoid 0.0333 0.9625
 Approximate Solution
Mode
Natural
Frequency
Damping
Ratio
Short Period 1.3328 0.3530
Phugoid 0.0452 0.7307
 Percentage Error in Approximate Solution
Mode
Natural
Frequency
Damping
Ratio
Short Period 0.96 0.70
Phugoid 35.73 24.08

16. Conclusion
 The phugoid approximations provide poor estimates while the short
period approximations are accurate.
 Time taken in short period mode to reach the equilibrium is 40-50
seconds after being disturbed and 280-300 seconds for phugoid
mode.