EGB211 – Computer Lab Assignment

Assessment No: 3

Assessment Type: Individual Computer Lab Assignment

Due Date: 11:59pm on Friday, 29th May 2020

EGB211 – Computer Lab Assignment

Introduction, Background and Project Overview

A dynamic problem can be split into two steps: (1) obtaining the equation of motion and (2) solving it.

To obtain the equation of motion we use common dynamics theory, such as Newton’s second law. The

equation of motion can then be solved through analytical (i.e. hand calculation) or numerical (i.e.

computational) techniques. So far, you have mostly looked at analytical solutions to problems, however,

even for simple dynamics problems, these can rapidly become complex and unwieldy, e.g., air

resistance (drag force). In reality, for many problems, the presence of a true analytical solution is rare

and instead solutions rely on simplifications and assumptions (e.g., small-angle theorem). On the other

hand, with numerical techniques solutions can be readily used to solve simple and complex dynamics

problems, without any simplifications. In numerical solutions, the main challenges become

computational resources and the elegance of the model (discretisation).

In this computer lab assignment, we will explore the analytical and numerical solution to the non-linear

equation of motion of the projectile illustrated in Figure 1. The projectile is an interesting problem

because when drag is ignored, an analytical solution is simple, however once drag is included the

problem becomes complex. This is due to the non-linear nature of drag. First, you will develop an

analytical solution, this can be achieved by neglecting drag. Next, you will explore the impact of drag,

which can not easily be solved analytically. You will then compare the two cases.

Figure 1. Projectile motion (not to scale)

Table 1. Quantities

Property Symbol Quantity Unit

Projectile

Mass ! 50e-3 "#

Radius $ 30e-3 !

Initial angle with respect

to the horizontal % −axis '( 25+ ,-#

Initial elevation ./ 50 !

Initial velocity |1/| 50 !. 345

Drag

Air density 6 1.183 "#.!47

Drag coefficient 89 0.80 -

For dynamic analysis of the projectile motion of the system illustrated in Figure 1 we will consider

the following:

• We will consider this as a two-dimensional problem in % and ..

• Unless in a complete vacuum, drag (air resistance) will act on the projectile.

The magnitude of the drag force is ||F||= 5; 896<‖1‖; and is always directed opposite the

projectile’s motion, i.e., the relative velocity vector (1) of the projectile.

• The initial speed ‖1/‖ does not result in the projectile breaking the sound barrier, that is, air

flowing over the projectile does not compress (>?@ℎ << 1) and, hence, 6 is constant.

• The vector components of the drag force F are identified by: DE = ||F||× cos('), where cos ' = LM‖1‖,

and, DN = ||F||× sin('), where sin ' = LQ‖1‖.

Since ||F|| is always positive its direction must be corrected to ensure that drag force vector F

is in the exact opposite direction of the projectile’s motion, i.e., the direction of the vector 1.

EGB211 – Computer Lab Assignment

1. Equation of Motion [2 Marks]

Develop the equations of motion of this projectile motion in Figure 1. Show all steps including

a free-body-diagram (include all drag force components).

Answer within this box. Change the size of the box if needed.

EGB211 – Computer Lab Assignment

2. Analytical Solution [2 Marks]

i. Linearization: Obtain the linearized version of the non-linear equations of motion

defined in part 1 by neglecting drag.

ii. Using this linearized equation of motion from (i), derive for the analytical solution for

the position, velocity and acceleration as functions of time.

iii. Determine the time needed for the projectile to reach the ground .(R) = 0 and the

corresponding distance travelled (%-coordinate).

Answer within this box. Change the size of the box if needed.

EGB211 – Computer Lab Assignment

3. Finite Differences [5 Marks]

i. Derive the finite difference equation for the projectile, starting from the non-linear

equation of motion obtained in part 1. Show all steps.

ii. In addition, define and justify how finite differences is initialised.

iii. Show your MATLAB code for the implementation of the finite difference problem

from (i) and (ii).

Answer within this box. Change the size of the box if needed.

(Include initialisation from (ii) and do not include the code part responsible for the plotting)

EGB211 – Computer Lab Assignment

4. Discretization and Convergence [5 Marks]

The resolution of your solution is completely dependent on time-step ,R and can increase

indefinitely as ,R → 0. Because of this, in the practical application of computational resources,

we are only interested in ,R which will provide results that do not change/benefit greatly from

any further increase in resolution.

Demonstrate and justify your choice for time-step ,R and the total simulation time. To start, set ,R = 1 [s] and choose a total time that allows for the project to reach the ground. Plot the

position (%, .) obtained from the Analytical and Numerical Solution neglecting drag force on

the same figure. (Note: one possible way to neglect drag force is by simply setting 8W = 0)

i. Comment on the resolution of results, is this ,R = 1 [s] adequate?

ii. If not, change ,R until sufficient resolution is achieved. To assess this, compare the

landing position %NX( for at least five different ,R against the analytical %NX( from

Part 2, using the table provided below. Show the plot of the position (%, .) with

your chosen value for ,R.

Answer within this box. Change the size of the box if needed.

Landing position YZX[ at various \]

Analytical %NX( = … [m]

Test No. ,R [s] Numerical (neglecting drag) [m] Error (%)

EGB211 – Computer Lab Assignment

5. Analytical vs. Numerical (neglecting drag force) [5 Marks]

Using the linear analytical solution from part 2 and the numerical solution from part 3

neglecting drag force (e.g., can be done by simply setting 8W = 0)

i. Show a separate plot for each of the following variables with respect to time: vertical

position, the magnitude of velocity, and magnitude of acceleration. In each plot, for

each variable, display both the Analytical and Numerical results for comparison.

[suggestion: use subplots to generate three separate plots for each variable all in one

figure. Plot from time R = 0 to the time taken to reach the ground, i.e. when RNX(.]

ii. Comment and provide an explanation for the agreement (or, disagreement) of

analytical vs. numerical solution (neglecting drag force) results.

Answer within this box. Change the size of the box if needed.

*Ensure your plots are labelled, easy to read, and well presented. omputer Lab Assignment

6. Numerical Investigation [6 Marks]

In this section, the effect of air resistance (drag force) will be investigated numerically by using

only the numerical solution (i.e., finite difference) from part 3 to solve the case where drag is

neglected (as done similarly in part 5) and the case where drag is included.

i. Plot the position (%, .) of the projectile for the results with and without drag force in

the same figure for comparison.

Also, show a separate plot for each of the following variables with respect to time:

vertical position, the magnitude of velocity, and magnitude of acceleration. In each

plot, for each variable, display both the results with and without force in the same

figure for comparison.

ii. Comment on the characteristics observed and provide an explanation for the

agreement (or, disagreement) of the numerical results with and without drag force.

Suggestion, consider the following:

• How well does the linear assumption compare against the fully non-linear

equation?

• Is drag relevant in the motion?

Answer within this box. Change the size of the box if needed.

*Ensure your plots are labelled, easy to read, and well presented.