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Homework answers / question archive / The assignment involves 3+2 tasks, where tasks 1, 2, and 3 require using Matlab software, and tasks 4 and 5 involve problem-solving and some minor Matlab-based (or any other comparable tool, e

The assignment involves 3+2 tasks, where tasks 1, 2, and 3 require using Matlab software, and tasks 4 and 5 involve problem-solving and some minor Matlab-based (or any other comparable tool, e.g. Python) calculations. The question will only run on Matlab by running >>vib2_tvalab_2022_full('dw20150') This assignment is due on the 8th of August and the question will be released on the 1st of August. (1 week time frame) Below is a very similar assignment.

Task 1: Single degree-of-freedom analysis (20% of 100%)

Specification: Use the provided Matlab app to find the physical parameters of the aerofoil (mass moment of inertia of the aerofoil about its hinge, torsional damping, torsional stiffness). Present the following Results and Discussion points:

• The main figure (graph) in Results should contain (1) the 1DOF FRF obtained using the harmonic excitation in the Matlab app, overlayed with (2) the FRF calculated analytically using the physical parameters, further combined with the information indicating (3) the 1DOF undamped natural frequency.

• The other shown Results should represent the summary table with the found parameters and one additional auxiliary figure illustrating (any) one of the methods used during parameter identification.

• One paragraph in your Discussion should summarise briefly the methods or approach used for parameter identification.

• One paragraph in your Discussion should briefly summarise the assumptions or criteria used during the parameter identification stage.

Guidance, suggestions, hints:

• Recommended layout: Results section 2/3 A4 page, Discussion 1/3 A4 page.

• The default system in the Matlab app, after its start, is 2DOF and needs to be modified to approximate the 1DOF aerofoil system by reducing, suitably, the significance of the TVA part in the

system.

• Use suitable input conditions (forced, free vibrations) to obtain such responses that can be used to calculate the parameters, e.g., LogDec, static or slow deformation, very high frequency excitation.

• For each FRF (or AfCh in some later tasks) use a suitable number of the frequency points to show the specifics of the function. Typically, between 10 and 20 points suffices for such a demonstration.

• Recommended resources: lectures 3, 4, 7, 8.

Task 2: General two degree-of-freedom analysis (20% of 100%)

Specification: Use the provided Matlab app to calculate 2DOF system AfCh and present the following Results

and Discussion points:

• The single main figure (graph) in Results should contain the following information identified with the

help of the Matlab app: (1) the initial 2DOF AfCh for the aerofoil DOF and (2) 2DOF AfCh for the TVA

DOF (calculated for the initial values of the aerofoil and TVA parameter values); (3) 1DOF AfCh for

the aerofoil DOF; (4) the information indicating the calculated 2DOF undamped natural frequencies

(e.g., labelled vertical lines).

• The other shown Results should represent the summary table with the values of the undamped

natural frequencies (also visualised in the main figure).

• One paragraph in your Discussion should comment briefly on the mutual relationship (i.e., similar or

dissimilar features) between the two calculated 2DOF AfCh functions for the aerofoil and TVA DOFs.

• One paragraph in your Discussion should comment briefly on the relationship (i.e., relative

placement) between the 2DOF AfCh function peaks and the two calculated undamped natural

frequencies.

• One paragraph in your Discussion should briefly discuss the role of the aerofoil and TVA damping on

the shape of the two AfCh functions.

Guidance, suggestions, hints:

• Recommended layout: Results section 2/3 A4 page, Discussion 1/3 A4 page

• Use your own initial (default) 2DOF system configuration in the Matlab app, in combination with the

suitable range of harmonic inputs, to calculate the amplitude frequency characteristics (AfCh

functions) such that both resonance regions are well described.

• Use the 1DOF approximation from Task 1 to calculate the corresponding 1DOF AfCh. Think about the

relationship between the AfCh (2DOF lectures) and FRF (1DOF lectures) functions.

o Note: based on how it is defined in the lecture notes, AfCh can be seen as a function that

presents the specific steady-state response to any form of the harmonic excitation. FRF is a

special function which presents the normalised steady-state harmonic response of an output

DOF to the harmonic load applied to the input DOF.

• Consider recalculating the relevant AfCh function(s) for the changed damping parameter value(s) in

order to support your discussion of the damping influence in the studied problem.

• To calculate the 2DOF undamped natural frequencies, you need to derive the EoMs for the 2DOF

problem (e.g., using Newton), determine the mass and stiffness matrices, and perform the

eigenvalue analysis.

• In your discussions, focus on similarities and differences, characteristic features, trends, root cause

of the behaviour of interest, significance, etc.

• Recommended resources: lectures 12, 14, 15, 16; recorded 2DOF (in-person) sessions.

Task 3: Tuned Vibration Absorber (20% of 100%)

Specification: Tune the TVA such that it absorbs or suppresses the resonance in your own 1DOF system.

Present the following Results and Discussion points:

• The first figure in Results should contain the following information identified with the help of the

Matlab app: (1) 1DOF AfCh (reused from Task 2); (2) 2DOF aerofoil AfCh for the case with the arbitrary

tuned TVA (i.e., without any additional constraints) and zero TVA damping; (3) 2DOF aerofoil AfCh

for the case where the TVA mass is 5% of the equivalent mass of the aerofoil (see hints) and 1% TVA

damping (see hints); (4) 2DOF aerofoil AfCh for the case where the TVA mass is 5% of the equivalent

mass of the aerofoil (see hints) and 20% TVA damping (see hints).

• The second figure in Results should compare the TVA configurations described above in (3) and (4)

by additionally including, in each case, the AfCh functions for the aerofoil DOF and TVA DOF.

• One paragraph in your Discussion should comment briefly on the characteristic features observed in

the first figure, particularly the effect of the TVA mass and TVA damping.

• One paragraph in your Discussion should comment briefly on the effect of the TVA damping on the

behaviour of the primary system (aerofoil) and secondary system (TVA) relative to each other.

Guidance, suggestions, hints:

• Recommended layout: Results section 2/3 A4 page, Discussion 1/3 A4 page.

• Use harmonic inputs and appropriate frequency domain sampling to recover the required AfCh

functions.

• To determine the equivalent mass of the aerofoil, use the concept of “radius of gyration” from

Mechanics formulated between the aerofoil hinge and the TVA attachment point.

o Note: mass moment of inertia = equivalent mass × (radius of gyration)2

, where “radius of

gyration” is known because we know where the “equivalent mass” is located for the purposes of this comparison.

• The “%-value TVA damping” represents the damping ratio of the TVA spring-damper-mass system when in isolation from the primary system (aerofoil) and acting as a 1DOF system.

• Recommended resources: lectures 15, 16; recorded 2DOF (in-person) sessions.

Task 4: Modelling 2 DOF system (20% of 100%)

Specification: Consider the 2DOF system below. With your own notation and for the wind off conditions,

derive the equations of motion using the Newton’s and Lagrange’s approaches.

Notes: This system represents a 2DOF pitching-heaving aeroelastic test rig. A pair of rigid plates is suspended on the

vertical springs. The plates are guided vertically without friction on the linear bearings (e1,e2). The aerofoil moves with

the plates and pivots about hinge (d) whilst being elastically suspended between point (b) and the plate. Point (a) is the

aerodynamic centre, (c) is the mass centre. The system should be modelled as a 2D problem as implied in the side view.

Where necessary, define your own parameters, coordinate system, coordinates, positive orientations, etc.

Present the following Results and Discussion points:

• In Results, include a picture of the system in the deformed configuration with the suitable coordinate

frame and positive orientations of the coordinates. Annotate the figure clearly.

• In Results, for the Newton’s approach: Provide separate and detailed free-body diagrams as well as

the corresponding equations of dynamic equilibrium.

• In Results, for the Lagrange’s approach: Provide all energy terms that constitute the Lagrangian

function and write down all terms of the Lagrange’s equations for the system.

• In Discussion, write down the equations of motion in matrix form obtained using both methods and

describe (compare) your results. In your descriptions, briefly compare the two sets of results and, if

not obvious from the figures above, describe all remaining quantities and parameters introduced

during the modelling.

Guidance, suggestions, hints:

• Recommended layout: Results section 3/4 A4 page, Discussion 1/4 A4 page

• Useful resources: lectures 11, 16, 17, 18; example sheets; recorded in-person sessions.

DOF 2 Rigid plate

Wall and rigid

column

Equilibrium

position

1.6

m

Linear

springs

3D view

Gravity

DOF 1

Aerofoil

a b c d

e1

e2

Side view

U=0 m/s

ρ=0 kg/m3

Task 5: Analysis of 2 DOF system (20% of 100%)

Specification: Perform eigenvalue and static aeroelastic analyses in Matlab on the 2DOF model derived in

Task 4. Specifically, calculate and plot the natural frequencies when varying the position of points (b)

between the leading edge and pivot point (d). Plot the corresponding mode shapes for one selected location

of point (b). Then, assume nonzero air speed and calculate the resulting steady-state (static) deformations in

the test rig relative to its equilibrium configuration.

Present the following Results and Discussion points:

• In Results, include one figure which shows the changes in the natural frequencies as a result of the

changing position of point (b) between the leading edge and pivot point (d).

• In Results, include one figure which shows (visualises) the mode shapes for one chosen aerofoil spring

placement.

• In Results, provide the full static aeroelastic equations, both as analytical expressions and their

numerical values. Provide the calculated values of the static deformations due to the air flow.

• In Discussion, briefly discuss and explain the trends in the natural frequencies when varying the

position of the aerofoil suspension point (b). Then, briefly describe the main features of the

presented mode shapes.

• In Discussion, explain the method or process used to determine the aerodynamic loads. After this,

briefly describe response of this rig to the loads arising from the static aeroelastic interaction.

Guidance, suggestions, hints:

• Recommended layout: Results section 3/4 A4 page, Discussion 1/4 A4 page

• Apply the Lagrange’s or Newton’s approach to determine the (generalised) loads due to aerodynamic

loading.

• Recommended resources: lectures 12, 17, 20, 21; “Matlab and vibrations - … revision note”; M-files

in “Matlab” folder.

Gravity

U≠0 m/s

ρ=1.225 kg/m3

b

d

Background and context

“Fighter aircraft have been designed to fly and maneuver at high angles of attack and at high loading

conditions. At these high angles of attack, the flow separates from the sharp leading edges of the wing and

leading-edge extension (LEX) forming a strong vortical flow that maintains the stability of the aircraft.

However, the leading-edge vortices break down upstream of the vertical tails. The breakdown flow impinges

upon the vertical tail surfaces causing severe structural fatigue and has lead to their premature fatigue failure,

costing millions of dollars every year for inspections and repairs.” [Sheta&Huttsell, 2003]

Moses, RW, Active vertical tail buffeting alleviation on a twin-tail fighter configuration in a wind tunnel, CEAS

International Forum on Aeroelasticity and Structural Dynamics. 1997.

Sheta, EF, Huttsell, LJ., Characteristics of F/A-18 vertical tail buffeting. Journal of fluids and structures, 2003

Mar 1;17(3):461-77.

Within the context of this coursework, you are a vibration specialist asked to analyse a problem of all-moving vertical stabiliser buffeting on a sixth-generation fighter aircraft. A range of tasks needs to be completed to characterise the system and propose the viable passive vibration control solution based on the application of the tuned vibration absorber concept.

The technical tasks have the following focus (detailed description follows later in this document):

1. Single DOF analysis using the provided Matlab model.

2. Two DOF analysis using the provided Matlab model and eigenvalue analysis.

3. Vibration absorber tuning combined with the use of the provided Matlab model.

4. Derivation of the equations of motion for a new two DOF system.

5. Analysis (eigenvalue, aeroelasticity) and Matlab work.

To complete the first three tasks, the provided Matlab application model (called “app”) will be used as a form

of a “digital twin” which represents the overall system with the unknown vertical stabiliser properties (later also called “wing” or “aerofoil”). The last two tasks are independent of the provided Matlab app

The tool and the modelled system

To support the first three tasks of this coursework, a simple Matlab app is provided in the P-file format to

avoid the necessity or possibility of the source code modification. The app represents a model of the

problematic vertical stabiliser with the unknown physical properties which will be determined in Task 1.

The system’s visual representation is shown in the following figure together with its constituent components,

model elements and relevant geometry information (note the offset dimensions).

This is a two degree-of-freedom problem where the first DOF is realised by the aerofoil pitching and the

second DOF is linked with the vertical motion of the TVA mass.

The application window is shown in the following figure where the main user blocks are identified and

described.

IMPORTANT: You must use the FULL version of the Matlab app which can be downloaded from “Assessment

Information” folder located in “Assessment, submission and feedback” section. Do not use the preliminary

version released for learning and demonstration purposes. In the FULL version, all students have their own

individual aerofoil parameters to work with. This capability is not present in the demo version.

Applied force F(t) Damped TVA m,c,k

Grounded torsional

spring kt and damper ct

Aerofoil/vertical stabiliser

(mass moment of inertia I

about the hinge)

b a

Pivot point/torsional

hinge

Studied system,

user inputs,

animation

Calculated

wing and TVA

response

Contextual information about the location of the evaluated system (all moving vertical USER

INPUTS

CALCULATED

OUTPUTS

User functionality

After each user-specified parameter update, the system’s vibration response is automatically updated and

shown in the bottom panel. The user inputs are applied in the top left panel. The top right panel serves for

simple contextual information about the overall system layout.

The user input windows can be activated by a mouse click on: “TVA” or the spring-mass visualisation of the

damped TVA (see figure below) to change the mass, damping and stiffness of the TVA; “Force” or the red

arrow visualisation of the input force to change the time-dependent excitation function (see the figure

below); “Solver” to adjust the ODE time integration range, discretisation and the initial conditions.

The time-dependent excitation function can be any valid Matlab expression with the specific parameter

values and a time symbol “t”. Examples of suitable functions are: “1” (a constant unit step at t=0 sec),

“30*sin(2*pi*3*t)” (a harmonic force at 3 Hz), “0” (a zero-force leading to free vibrations for the non-zero

initial conditions), “0.1*t” (a slow ramp function). More complicated function specifications are possible. The

four initial conditions represent: (1) initial aerofoil angular displacement in [rad]; (2) vertical TVA

displacement in [m]; (3) aerofoil angular velocity in [rad/s]; (4) vertical TVA velocity in [m/s].

For precise analysis purposes, the calculated vibration responses can be plotted separately after a mouse

click on any of these lines. A separate window will be generated which will show the applied force in [N] (the

top subplot). The bottom subplot contains the aerofoil angular displacement (shown in blue) in [rad], and

the vertical TVA mass coordinate (shown in green) in [m].

A mouse click on one of the calculated vibration response curves also updates the content of the variable

“var_tvalab” which is present in the base workspace (i.e., through command line) and can be used for

further analysis in or outside of Matlab. The columns in this matrix represent [time,dof1,dof2,force] and the

rows represent the time instants for which the responses were calculated.

An animation of the obtained responses can be started (stopped) by clicking on “Play” (“Stop”) text fields. A

click on “Legend” text field switches on/off a legend in the calculated vibration response panel.

Specify TVA parameters Specify excitation function Specify ODE integration options

Excitation force [N]

Blue: wing (aerofoil)

pitch angle [rad]

Green: TVA mass

displacement [m]

App start and initial configuration

Each student must download their own FULL version from the “Assessment Information” folder located in

the “Assessment, submission and feedback” section of the unit to be able to work with their own problem

specification. In other words, every student has their own specific aerofoil parameter values to determine in

Task 1.

The FULL version of Matlab app must be started using a single input argument which represents the student’s

“UOB USER NAME” typed in lower case on the Matlab command line. The P-file must be located in the

directory visible to Matlab, e.g., current working directory.

Example:

vib2_tvalab_2021_full('js00000') % use your own UOB USER NAME

After this, the application starts and is ready to be used for further analysis:

The default initial state of the app is represented by the student’s specific aerofoil parameters, the identical

and adjustable TVA parameters, and semi-randomised harmonic excitation force. The force and TVA

parameters can be changed. The aerofoil parameters are fixed and specific to each student.

Final remarks: It is recommended to run the app on your own machines, assuming you have a Matlab

installation. However, in case of problems of any kind on your local machines, you can work with this app on

your own UOB remote desktops. This option was assessed and was found to be a feasible option for the use

of this tool. This Matlab app does not assume or require the use of any special toolbox, only the core Matlab

functionality is used.

End of the document.

Initiate Matlab app to its default

configuration specific to each student

using the FULL version and your own

UOB USER NAME.

Click on the calculated response time

series to open results in a separate

window for further analysis.