Engineering Modeling and Design: Missile Aerodynamic Drag Simulation

This project focuses on the numerical simulation of a missile subjected to aerodynamic drag. It begins with an introduction explaining the complexity of modeling the motion of objects in fluids and identifies the forces involved. The project then presents the mathematical model, including the equations of motion in both horizontal and vertical directions, and defines the constants used. The analytical solution involves deriving the state-space representation. The core of the project is the MATLAB solution, using ODE45 to solve the equations numerically and plotting the flight path and velocity over time. Assumptions for initial conditions and specific scenarios are outlined, and the MATLAB code is provided. The results include plots demonstrating the missile's trajectory under various conditions. The project concludes with a discussion of lessons learned, emphasizing the relationship between different parameters and their effect on flight patterns. It also suggests future work to incorporate additional forces for a more realistic simulation. References to relevant research papers are included.

Mechanical Engg.
MCEG 3003
Engineering Modeling and Design
Numerical Simulation of a Missile subjected to Aerodynamics Drag
Student Name:
Percentage Contribution of student:

Introduction
The aim of the project is to get the numerical solution of a missile which is subject to
aerodynamic drag during its motion. The mathematical models describing the equations of
motion of the missile are mentioned.
The motion of bodies in a fluid is a very complex engineering problem and its not totally
understandable even till today. The reason that this problem is so complex is that many types
of forces act on a body in a fluid. The various forces are : Gravity, Lift, Thrust, Drag,
Buoyancy, and, Inertial forces as per Pawar (2018) and Sun (2019).
In the project, the model of a missile subject to only aerodynamic drag is designed.
Methodology :
A simple mathematical model describing the equations of motion has been mentioned as
follows:
𝑚 𝑥̈ + 𝑐𝑑 𝑥̇ ( 𝑥̇ 2 + 𝑦̇ 2 ) 1 / 2 = 0 ( 1 )
𝑚 𝑦̈ + 𝑐𝑑 𝑦̇ ( 𝑥̇ 2 + 𝑦̇ 2 )1 / 2 = − 𝑚𝑔 ( 2 )
The model consists of the terms related to ‘ x ’ and ‘ y ’. Here, ‘ x ’ denotes the horizontal
displacement, ‘ y ’ denotes the vertical displacement. ‘ 𝑥̇ ’ denotes the horizontal velocity,
‘𝑦̇ ’ denotes the vertical velocity, ‘ 𝑥̈ ’ denotes the horizontal acceleration and ‘ 𝑦̈ ’ denotes
the vertical acceleration. Some constants are also used here like ‘ m ’, ‘ cd ’ and ‘ g ’.
Here, m – Missile mass = 800 ( kg )
x – Spatial horizontal distance travelled
y – Spatial vertical distance travelled
𝑐𝑑 – Aerodynamic drag coefficient = 0.15 ( - )

g – Gravitational acceleration = 9.81 (m/s^2)

When the system of ODEs is solved, the distance and velocity of the object in horizontal and
vertical directions can be obtained as results.
Analytical solutions:
Derivation of the state-space representation for given problem :
The state space representation is as follows as per Pawar ( 2019 ) and Szmuk ( 2016 ) .
Adding both the equations:
𝑚 𝑥̈ + 𝑐𝑑 𝑥̇ ( 𝑥̇ 2 + 𝑦̇ 2 ) 1 / 2 + 𝑚 𝑦̈ + 𝑐𝑑 𝑦̇ ( 𝑥̇ 2 + 𝑦̇ 2 )1 / 2 = − 𝑚𝑔
𝑚 𝑥̈ + 𝑚 𝑦̈ + 𝑐𝑑 𝑥̇ ( 𝑥̇ 2 + 𝑦̇ 2 ) 1 / 2 + 𝑐𝑑 𝑦̇ ( 𝑥̇ 2 + 𝑦̇ 2 )1 / 2 = − 𝑚𝑔
𝑚 ( 𝑥̈ + 𝑦̈ ) + 𝑐𝑑 ( 𝑥̇ 2 + 𝑦̇ 2 ) 1 / 2 ( 𝑥̇ + 𝑦̇ ) = − 𝑚𝑔
𝑚 ( a ) + 𝑐𝑑 | v | ( v ) = − 𝑚𝑔
Here a = 𝑥̈ + 𝑦̈ , v = 𝑥̇ + 𝑦̇ , | v | = ( 𝑥̇ 2 + 𝑦̇ 2 ) 1 / 2
. a = dv / dt
𝑚 dv / dt + 𝑐𝑑 | v | ( v ) = − 𝑚𝑔
This is a reduced differential equation.

MATLAB solutions:
Assumptions:
Assume the initial condition to be 𝑥(0)=𝑦(0)=0, 𝑥̇ (0)=58 (𝑚𝑠) and 𝑦̇ (0)=32 (𝑚𝑠)
MATLAB Solution
MATLAB (ODE23 or ODE45) is used for solving the equations (1) and (2) numerically and
plotting the distance (flight path) and the velocity of the object in time. We have used ODE
45.
Code:
syms x ( t ) y ( t )
t0 = 0 ;
tf = 100000 ;
dx = diff ( x , t ) ;
dy = diff ( y , t );
c1 = 0 ;
c2 = 0 ;
c3 = 58 ;
c4 = 32 ;
y0 = [ c1 c2 c3 c4 ] ;
eq1 = 800 * diff ( x , 2 ) = = -0.15 * dx * sqrt ( dx * dx + dy * dy ) ;
eq2 = 800 * diff( y , 2 ) + 800 * 9.81 = = - 0.15 * dy * sqrt ( dx * dx +
dy * dy ) ;
vars = [ x ( t ); y ( t ) ]
V = odeToVectorField ( [ eq1 , eq2 ] )
M = matlabFunction( V ,' vars ', { ' t ' , ' Y ' } ) ;
interval = [ t0 tf ] ;
ySol = ode45 ( M , interval , y0 ) ;
tValues = linspace ( interval ( 1 ) , interval ( 2 ) , 1000 ) ;
yValues = deval ( ySol , tValues , 1 ) ;
plot ( tValues , yValues )

Plot:

Initial conditions that make the object to travel the longest horizontal distance ( assuming that
the total travelling time should not be more than 60 seconds)
Initial velocity = 900 g = 8829 m/s
The travelled distance and velocity of above part after 25 seconds.
Angle of departure the object will have in above part.
Angle = 45 degree
Conclusion:
Lesson learn from the project
The designing process helps to understand the way different parameters and initial conditions
are related to each other and the way they affect the flight pattern. Here, only drag force is
considered. Other types of forces have been neglected.
Future work to make this work more realistic
In future work, the effect of more number of forces can be included in the design equations.
This will make the work more realistic as per Ge (2017) ,Xu (2019), Mumivand (2016) and
Zhu (2018).
References
Pawar, R.S., Gilke, N.R. and Warade, V.P., 2018, July. Numerical Simulation Over Flat-Disk
Aerospike at Mach 6. In 2018 IEEE International Conference on System,
Computation, Automation and Networking (ICSCA) (pp. 1-6). IEEE.

Sun, X., Huang, W., Ou, M., Zhang, R. and Li, S., 2019. A survey on numerical simulations
of drag and heat reduction mechanism in supersonic/hypersonic flows. Chinese
Journal of Aeronautics.
Pawar, R.S., Gilke, N.R. and Warade, V.P., 2019. Numerical Simulation Over Conical
Aerospike at Mach 6. In Proceedings of International Conference on Intelligent
Manufacturing and Automation (pp. 15-26). Springer, Singapore.
Szmuk, M., Acikmese, B. and Berning, A.W., 2016. Successive convexification for fuel-
optimal powered landing with aerodynamic drag and non-convex constraints. In AIAA
guidance, navigation, and control conference (p. 0378).
Ge, Z., Wang, Y., Huang, Y. and Liu, S., 2017, August. Segmented optimal design of
ballistic trajectory of gliding extended range projectile subjected to multiple
constraints. In 2017 International Conference on Robotics and Automation Sciences
(ICRAS) (pp. 108-113). IEEE.
Xu, X., Wei, Z., Ren, Z. and Li, S., 2019. Fault-Tolerant Time-Varying Formation Tracking
for Second-Order Multi-agent Systems Subjected to Directed Topologies and
Actuator Failures with Application to Cruise Missiles. In Proceedings of 2018
Chinese Intelligent Systems Conference (pp. 845-861). Springer, Singapore.
Mumivand, M. and Mohammadkhani, H., 2016. Numerical study of aerodynamic drag
reduction of blunt nose with hybrid of spike and axial and lateral jet
injection. Modares Mechanical Engineering, 16(7), pp.133-142.
Zhu, S., Chen, Z., Zhang, H., Huang, Z. and Zhang, H., 2018. Investigations on the influence
of control devices to the separation characteristics of a missile from the internal
weapons bay. Journal of Mechanical Science and Technology, 32(5), pp.2047-2057.