代写EMS726U/P Engineering Design Optimisation and Decision Making CW2 - Paper Helicopter代做留学生SQL语言程序

EMS726U/P

Engineering Design Optimisation and Decision Making

CW2 - Paper Helicopter:

Multi-objective Experiment Optimum Engineering Design

2024/2025

1.   Problem description

In the helicopter design industry, autorotation is the rotation of helicopter rotors in response to downward motion when no power is available, it is critical to the survival of pilots and passengers when a helicopter is in an emergency engine-out condition. To maximise the effectiveness of autorotation, manufacturer requires a large investment of intellectual and material resources, which increases the costs of assembling. That is, managing the autorotation effectiveness is weighed against controlling the assembling costs. This coursework requires participants to formulate a multi-objective engineering design problem, using a paper helicopter as a simplified and abstractive model to get insights of the tradeoff between maximising autorotation effectiveness and minimising assembling costs. The autorotation effectiveness is measured by the time it takes a paper helicopter to reach the ground from an initial altitude (~2.5 meters),  i. e., the longer the time, the better the effectiveness. The assembling cost corresponds to the entire area of the paper helicopter, the larger the entire area, the higher the cost.

A paper helicopter is made from a sheet of A4 size paper (210mm × 297mm) and consists of two rotors, one body, one tail, and one paper clip placed on the bottom of the tail. Paper clip plays a role as the stabiliser of the paper helicopter. Figures 1 and 2 show the 2D and 3D templates. The length of rotors and the tail area (both tail height and width) are three independent variables that need to be optimised.

Figure 1 Paper helicopter 2D template Figure 2 Paper helicopter 3D template

There are 7 tasks to be completed in order to deliver a group report and a group presentation (see Section 4):

1. Mathematical analysis on the relationship between the independent variables and optimisation objectives. Based on the equations given in Section 2, please derive two equations showing how the autorotation effectiveness and assembling cost are affected by the independent variables respectively.

2. Basic data collection. Please first check the necessary basic information listed in Table 1, then search and fill in the missing parameter values using information available online or from industrial standards.

3. Paper helicopter design. Use the derived equations obtained from Task 1 and parameters from Task 2 to model the multi-objective paper helicopter optimisation problem in Pymoo (covered in the IT class of Week 9). Choose one of the multi-objective optimisation algorithms from Pymoo to design the helicopter. Justify your choice.

4. Paper helicopter making. According to the configurations obtained from Task 3, use the weighted decision matrix method to select several configurations from the Pareto front. Select and justify appropriate weights to obtain configurations from the desired region of the Pareto front. For each configuration selected, please make 2 paper helicopters, where one for tests and the other for Task 6.

5. Performance testing. The test priority is to check if the designed configurations can make paper helicopters rotate  effectively  during  the  whole  falling,  rather  than  drifting  down  and  swaying  around.  If some configurations fail to reach this goal, please analyse the reason, and then go back to Tasks 3 and 4.

6. Experiments.  Use paper helicopters with configurations  that have passed the tests in Task 5, record performance values on two objectives (when counting the rotating time, please carry out at least 5 runs for each configuration and calculate the average) and take one video of each configuration experiment.

7. Write a detailed report and prepare slides for the presentation based on the results of Tasks 1~6. Please refer to Section 4 Assessment for more information.

2.   Analytical design

The downward rotation process of the paper helicopter is almost a uniform linear motion, where the upward drag force is balanced with the downward gravity.

G = D = 2/1pairV2 CDS                                                                            (1)

where  G  and  D   denote the entire gravity and drag respectively,  pair    is the air density,  V   is the downward velocity,  CD    denotes the air drag coefficient, and  S  is the area spanned by the rotors and S = πLR2 , in which LR   is the length of rotor.

Therefore, the steady-state velocity  V   is obtained via Eq.2.

(2)

This  steady-state velocity  V   can be used as  a  surrogate  objective,  since  minimising  it  is  equivalent to maximising the descent time,  i. e., maximising the autorotation effectiveness. When doing Tasks 1, 3 and 5 in Section 1, please use minimising the steady-state velocity and minimising the assembling costs as the objectives, in order to make the Pareto front a convex curve.

The entire gravity  G   is the sum of the gravities of the rotors, body, tail, and paperclip, as shown in Eq.3.

G = Grotors  + Gbody  + G tail  + Gclip                                                                                     (3)

The gravities of the body and tail are calculated by multiplying their masses by gravitational acceleration, where each mass is determined by multiplying its area by the paper density. The gravity of rotors is assumed to increase as the cube of its radius, to reflect strength and stiffness requirements in a real helicopter.

Grotors  = Grotors_0 (LR/LR_0 )3                                                                                              (4) 

where  Grotors_0   is the initial gravity of rotors with the initial length  LR_0 . The value of LR_0    is in Table 1.

Altogether, the gravity in Eq.3 can be expressed in Eq.5, where  W   and  H  denote the values of width and height respectively, with  B  and  T  indicating the body and tail respectively.

G = Grotors_0 (LR/LR_0 )3 + WBHBppaperg + WTHTppaperg + Gclip                                             (5)

By Eq.5, Eq.2 can be written in an equivalent form in Eq.6.

V2 LR2  = f1 (LR, WT, HT) = a1LR3  + a2WTHT  + a3                                                                   (6)

Please derive the constants  a1, a2   and  a3    in Eq.6 to complete Task 1 in Section 1. Note that, the final results of Task 1 should be in the form of Eqs.7 and 8 below.

V = f2 (∙) = ⋯                                                                                   (7)

Cost = f3 (∙) = ⋯                                                                                  (8)

Please note f1, f2   and  f3    in Eqs.6-8 denote functions, instead of implying there three objective functions.

Table 1 Experiment parameters

Variable	Description	Value
WR	The width of one rotor	6 cm
LR_0	The initial length of the rotors	8 cm
LR	The length of the rotors, to be optimised	8 cm ≤ LR ≤ 15 cm
WB	The width of the body	12 cm
HB	The height of the body	4 cm
WT	The width of the tail, to be optimised	2 cm ≤ WT ≤ 5 cm
HT	The height of the tail, to be optimised	5 cm ≤ HT ≤ 10.6 cm
pair	Air density	To be searched and filled in
CD	Air drag coefficient
g	Gravitational acceleration
ppaper	A4 paper density in experiments
Gclip	The gravity of one paperclip in experiments
Grotors_0	The gravity of rotors with the initial length LR_0	To be calculated

3. Safety notice

The safety of all participants is uppermost. The potential danger mainly involves:

Sharp tools that maybe used during hand-making process, like scissors and knife to cut papers.

Risk of falling from height. Tasks 5 and 6 in Section 1 need to release paper helicopters from a height of

2.5 meters. These two tasks can be either completed on campus or somewhere safe agreed by all group members. If participants would like to do the two tasks on campus, the Whitehead Aeronautical Laboratory is available from 13:00 to 14:00 on Monday 9 Dec. and from 15:00 to 16:00 on Wednesday 11 Dec. (Week 12). If preferring somewhere else, please be aware of the potential risks as mentioned above during the whole test/experiment process.

Whitehead Aeronautical Laboratory address: ENG-110, Engineering Building (east part), Mile End Campus.  Note that, the time slot of Whitehead Aeronautical Laboratory is only for you to carry out experiments by yourselves. One demonstrator will be there to maintain the order and ensure safety, but no coursework-related questions will be answered. Should participants have any questions, please make sure that you attend the Week 10 Problem-Solving Class. Please get all necessary materials ready before visiting Whitehead Aeronautical Laboratory for tests and experiments to be efficient and productive.