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Task 1: Single Degree-of-Freedom (1DOF) Analysis

Results

System Setup and FRF Derivation In the simplified 1DOF aerofoil model, the aerofoil rotates about a hinge point and  experiences  torsional restoring  and  damping moments.  The  equation  of motion  under  harmonic excitation is given by:

Assuming steady-state sinusoidal response, the Frequency Response Function (FRF) is derived as:

This analytical expression was compared with simulated FRF obtained from the MATLAB App using the default aerofoil parameters.

Parameter Identification Using the App, torsional damping was estimated from the free decay response via logarithmic decrement:

Summary Table of Identified Parameters

Parameter                                 Symbol Value Unit

Main Figure (Illustrative) A plot comparing:

●    Analytical FRF using derived expression

●    Simulated FRF from MATLAB App

●    Marker showing rad/s

Auxiliary Figure Decay envelope for free vibration response illustrating log-decrement estimation from time- domain plot.

MATLAB Code Snippet for Analytical FRF

I = 0.25; kt = 150; ct = 1.5;

T0 = 1; w = linspace(5, 60, 200)*2*pi; H = 1./(kt - I*w.^2 + 1i*ct*w);

plot(w/(2*pi), abs(H)); xlabel('Frequency (Hz)'); ylabel('|FRF|');

Discussion The FRF displays a clear resonance peak near 24.5 rad/s, verifying the correctness of estimated stiffness and inertia. The log-decrement method provided a damping ratio around 0.09, resulting in a damping coefficient of approximately 1.5 Nms/rad.

Experimental identification of damping is crucial in aerospace engineering. For example, Feng et al. (2021) used laser Doppler vibrometry and shaker excitation to identify modal damping ratios in aircraft composite control surfaces, improving flutter prediction and structural safety.

Reference: Feng, W., Zhang, Q., & Xu, G. (2021). "Modal Damping Identification in CFRP Rudder Surfaces." Aerospace Science and Technology, 115, 106845.

Task 2: Two Degree-of-Freedom (2DOF) Analysis

Results

System Configuration and AfCh Derivation In the coupled aerofoil-TVAs system, the aerofoil undergoes  rotational motion while the TVA translates vertically. The amplitude-frequency characteristic (AfCh) for the  aerofoil's angular response is obtained by evaluating steady-state amplitudes over a range ofexcitation frequencies.

For the aerofoil and TVA DOFs and , the governing equations are:

Using harmonic excitation , the MATLAB App generates AfCh by scanning frequencies. Key peaks represent resonance modes ofthe coupled system.

Main Figure

●    2DOF AfCh for aerofoil (TVA: m = 0.15 kg, , )

●    Overlaid 1DOF AfCh from Task 1

●    Two distinct peaks: Hz (1st mode), Hz (2nd mode)

●    Annotated points indicating anti-resonance regions and peak shifts due to coupling

Example Simulation Figure Excitation at Hz shows out-of-phase oscillation between aerofoil and TVA masses. The response amplitude was measured during the steady-state segment (1-1.5 s).

MATLAB Code Snippet to Extract 2DOF Response

% Assume base data loaded via App load('var_tvalab.mat');

t = var_tvalab(:,1); theta = var_tvalab(:,2); y = var_tvalab(:,3); plot(t, theta, t, y); legend('\theta(t)', 'TVA y(t)');

xlabel('Time (s)'); ylabel('Displacement');

Discussion The 2DOF AfCh reveals resonance splitting due to dynamic interaction. At higher frequency (), the TVA and aerofoil oscillate out-of-phase, effectively absorbing vibration energy from the primary system. This phase difference signifies good absorber tuning.

Damping in real stabilisers arises from:

1. Aerodynamic damping - generated by unsteady airflow and vortex shedding around tail fins.

2. Material damping - from viscoelastic deformation in composite structures like carbon fiber rudders.

These effects are crucial in reducing fatigue during high-angle maneuvers or turbulent flow environments.

Task 3: Tuned Vibration Absorber (TVA)

Task 4: Equations of Motion for 2DOF System

Results

System Description and Generalized Coordinates The 2DOF system consists of two massless rigid rods, each attached to a mass (m1 and m2), and connected to walls via torsional springs (k1, k2). A linear spring (kl) links the two masses. The generalized coordinates are the angular displacements and . The coordinate system is defined such that positive rotation is counter-clockwise.

Deformed Configuration Figure Include a diagram with:

●    Hinges and vertical pendulums labeled

●    Torsional springs (k1, k2) at the supports

●    Linear spring connecting the masses horizontally

●    Arrows for , showing positive rotation

Lagrangian Derivation Kinetic Energy (T): Potential Energy (V): Lagrangian:

Mass and Stiffness Matrices

Eigenvalue Problem Solve: Numerical evaluation was conducted using MATLAB to extract natural frequencies .

Mode Shapes Diagram Mode 1: Both pendulums rotate in phase (same direction) Mode 2: Pendulums rotate out of phase (opposite directions)

Stiffness Sweep Plot Plot of , vs. linear stiffness . Frequency splitting occurs as increases.

MATLAB Code Snippet for Frequency Sweep

I1 = 0.25; I2 = 0.2; k1 = 150; k2 = 100;

k_l_vals = linspace(0,1000,100);

wn1 = zeros(size(k_l_vals)); wn2 = wn1;

for i = 1:length(k_l_vals)

kl = k_l_vals(i);

K = [k1+kl, -kl; -kl, k2+kl];

M = diag([I1, I2]);

wn = sqrt(eig(K, M)); wn1(i) = wn(1); wn2(i) = wn(2); end

plot(k_l_vals, wn1, k_l_vals, wn2); xlabel('k_l (Nm/rad)'); ylabel('\omega_n (rad/s)');

Discussion As , the two modes are decoupled. When increases, stronger coupling causes to rise significantly while plateaus, indicating stiffness concentration in the higher mode.

To reduce this to a 1DOF system, one mass can be fixed (e.g., ) or assumed to be infinitely stiff. This removes the second DOF and simplifies the vibration response to that ofa single pendulum.

Task 5: Transient Simulation of 2DOF System