How is in a conventional airplane the longitudinal stability achieved?
4.1 What problems with longitudinal stability arise when designing a flying wing?
4.2 How can these problems be solved? Focus especially on different kind of aerofoils. Also make a free body diagram to show why certain aerofoils can’t provide longitudinal stability and others can.
4.3 Can a tailless aircraft be made longitudinally stable by applying sweep? Explain why or why not.
4.4 How is a canard used to achieve longitudinal stability in a delta wing aircraft like the Eurofighter Typhoon (Figure 6)?
4.5 Draw a swept flying wing in a (3-dimensional) body fixed reference frame and indicate the following points
Centre of gravity
Aerodynamic centre
Centre of pressure
Also draw the weight- and lift forces and the moment coefficient around the y-axis at the correct location. Use this figure to indicate how longitudinal stability is achieved.
Lift, weight, drag and lateral stability have been explained and questions have been answered about these topics. The only force yet to explore is the thrust force. This will be explored in the next chapter, closing the sequence of chapters on forces.
Figure 6: The Eurofighter Typhoon with its delta wing and full movable canards (www.eurofighter.com)
37 TH Thrust
Your flying wing uses an electrically driven propeller for thrust and control. The amount of thrust and power the propeller generates depends on the airspeed, the rotational speed, the size and pitch of the blade.
For this practical the group will perform wind tunnel measurements on propellers powered by an electrical engine, in order to create performance diagrams of the propellers. Additionally, the measurement results will be compared to theoretical results which are obtained using the blade element method, combined with momentum theory.
Learning objectives
The student should be able to:
Know the following parameters; Power available (Pa), Shaft power (Pbr) and propulsive efficiency.
Understand the relations between Thrust, propulsive efficiency, Pa, Pbr and the airspeed.
Gain understanding in the effects of changes in key propeller design parameters, such as blade pitch and rotational speed, on the performance of propellers.
Use blade element theory to perform basic performance calculations on propellers.
Schedule
(15 min) Preparation and familiarizing with the theory. (15 min) Divide the tasks over the group
(90 min) Do measurements in the wind tunnel.
(90 min) Work out the exercises (could be before tunnel).
(30 min) Discuss results.
Deliverables
Make sure that you write clearly on everything you hand in who has worked on it. Propeller performance diagrams o theory and
o measurement results
Answers to the questions of Task 3.
Clear table containing work and hour distributions of all team-members.
Tasks
Please follow the instructions of the student-assistant/mentor during the wind tunnel measurements. For the theoretical analysis, a brief explanation of the blade element method is given below. However, you are encouraged to look up the given reference, or any other source of information for more detailed information.
Task 1: Wind tunnel measurement
You are asked to produce: (1) Thrust vs. Velocity, (2) Power available vs. Velocity, and (3) Propeller efficiency vs. Velocity graphs for the combination of propeller and motor used in the test set-up. To do so, you can either record the values of the different variables by hand, or input them directly into a program like Microsoft Excel, if you have a laptop available. The plots are to be created for different engine settings, e.g. 20%, 40%, 60%, 80% and 100% of maximum shaft rotational speed.
The parameters measured are the velocity [m/s], the thrust [g], the voltage [V], the current [A]
and the propeller speed [rpm]. Please keep in mind that some of the parameters need to be converted to the correct units for some of the calculations.
The propulsive efficiency of a propeller is defined as the ratio of power available over shaft power (Ruijgrok, 2009).
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In which Pbr can be calculated using:
P
br M
(2)Where M is the torque [Nm] applied to the propeller and ω is the shaft speed [rad/s]. For the motor used, the output torque can be approximated by rewriting the following formula (Maxon Motor, 2008), see also Figure 1.
Figure 1: Speed-Torque line provided by Maxon Motor (2008)
Where n and n0 are the shaft speed and the shaft speed at zero torque, respectively. Their units are in [rpm]. The units of
n
M
are [rpm/Nm] and those of M are in [Nm] respectively.Furthermore, the following relation is given (Maxon motor, 2008):
0 n
n U k
(4)Here, U [V] is the applied voltage and kn [rpm/V] is the speed constant.
Data for the motor, provided by the manufacturer, is summarized in Table 1, including the constants needed in the previous equations.
Table 1: Motor data, supplied by Manufacturer
Maxon Motor RE 310007
Max permissible speed 12000 [rpm]
Stall torque 1020 [Nm]
Speed constant 369 [rpm/V]
n M
8.69x103 [rpm/Nm]As a final step, you are asked to calculate the maximum possible theoretical propeller efficiency, by using the measured thrust. To do this you can use the momentum theory, as explained in
Torque M
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(Ruijgrok, 2009, p140-143). To obtain a value for the density, you can assume ISA conditions along with the assumption that the wind tunnel is situated at sea level.
Task 2: Theoretical Calculations
The method you are going to use for the theoretical calculations is the blade element theory, combined with momentum theory, as explained in (Ruijgrok, 2009, p140-143). The blade element method is a numerical integration technique; meaning that in order to calculate the thrust (=component of the lift) generated by a propeller blade, the propeller blade will be divided into several smaller sections; each considered to have its own constant wing section.
Lift and drag of each element are calculated using two-dimensional aerodynamics. Next, the lift and drag of all blade elements, are summed up (numerical integration) to obtain the thrust and torque of the complete propeller blade. So instead of performing calculations on the entire blade, which has variable chord lengths, variable twist, variable wing profiles, etc; we simplify the blade by assuming that it can be modelled by several separate sections with constant parameters. Of course, this means that we are approximating the actual thrust and that there will be a difference between the calculated and actual thrust. This difference can be reduced by increasing the number of sections. More on numerical integration techniques will follow in 2nd and 3rd year courses. For more information on the blade element method and momentum theory, please consult the book of Ruijgrok.
The data needed for this task can be found on BrightSpace in a file called slow_fly1.xls and slow_fly2.xls. If your group is performing the theoretical calculations before doing the wind tunnel measurement, use slow_fly1.xls. Alternatively, if your group has already done the wind tunnel measurement, use slow_fly2.xls.
Figure 2: Propeller blade section (Ruijgrok, 2009)
To begin with the blade element method, the propeller blade should be divided into separate sections, as shown in Figure 2. The section location and width can be found in the excel file.
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Figure 3: Section Angles, Velocities and Forces (Ruijgrok, 1996)
Figure 3 gives an overview of the angles, velocities and forces for a section. It is possible to calculate the induced velocity Vi by using momentum theory. However, the induced velocity is a function of the thrust of the propeller whilst the thrust depends on the induced velocity. The induced velocity can therefore only be calculated through an iterative process. For this task, you can assume that the induced velocity Vi is equal to zero. Thus, the angle of attack α, is equal to the geometric pitch angle β minus the ‘flight path’ angle Φ. Please also note that usually the Greek letter Φ is used for the pitch angle.
Calculate, for each section, the angular velocity (R), and the total velocity (Vr) for the combinations of rpm and flight speed given in Table 2. As a next step, calculate the ‘flight path’
angle and the angle of attack.
Table 2: Combinations of rpm and flight speed Rpm [-] Flight speed [m/s]
8000 10 8000 15 8000 20 10000 10 10000 15 10000 20
Now that you know the angle of attack, you can use the CL and CD data given in the excel file to calculate the lift and drag for each aerofoil section. You may need to use interpolation to find a correct value for the calculated angle of attack. Knowing CL and CD, you can now calculate dL and dD for the sections by applying the standard lift and drag formulas to each section.
1
2 L2
L c V S
(4)1
2 D2
D c V S
(5)Here, S is the area of the section. Now calculate dT and dK for each section and then calculate the thrust and torque produced by the entire propeller for all combinations of the flight speed and rpm. Please note that you only calculated the contribution of one propeller blade. Can you now predict how the thrust varies with flight speed and rpm?
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On BrightSpace you will also find several Excel sheets with names starting with ThrustCalc. You can use these to check if your prediction was correct. Please note that this program uses an iterative process to calculate the induced velocity Vi.
Task 3: Compare results
Each group has measured different propellers during today’s wind tunnel test and subsequently has different plots. Today’s final assignment is to present your plots to the other groups and compare your results with theirs. Answer the following questions:
How do the theoretical results compare to the wind tunnel measurements? Can you explain the differences?
What is the effect of changing the propeller pitch?
Does changing the diameter of the propeller have any effect?
Can you explain what the benefit of a variable pitch propeller is?
What is the main reason why variable pitch propellers are not always used in propeller aircraft? Can you think of another reason to use a fixed pitch propeller?