Lab 02: Flexible Link Modeling | Anuranan Bharadwaj

Lab Overview

Course: AE 443 – Experimental Dynamics and Control Laboratory (Bernardo Restrepo / Jared Getz) | February 27, 2026
Objective: Derive the equations of motion for the SRV02 rotary flexible link system via the Euler-Lagrange method, experimentally determine the link's natural frequency and stiffness from a free-oscillation response, and validate a 4-state state-space model against measured hardware data.
System states: x = [θ, α, θ̇, α̇]ᵀ where θ is the servo angle and α is the flexible link deflection angle
Hardware: SRV02 with rotary flexible link, strain gauge sensor, Q2-USB DAQ
Parameters: B_eq = 0.004 N·m/(rad/s), J_eq = 2.08×10⁻³ kg·m², m_l = 0.065 kg, L_l = 0.419 m

SRV02 rotary flexible link system

Free-oscillation response (Fig. 1)

Free-oscillation with data point cursors (Fig. 4)

Equations of Motion & State-Space Model

The system Lagrangian L = T − V was formed using the kinetic energy T = ½J_eqθ̇² + ½J_l(θ̇ + α̇)² and elastic potential V = ½K_sα². Applying the Euler-Lagrange equations yielded two coupled equations of motion — a servo equation and a link equation. Solving these with B_l = 0 (undamped link assumption) produced the 4×4 state matrix A and 4×1 input matrix B.

Natural frequency was extracted from the free-oscillation response using three successive peak amplitudes to compute the period T_osc, the damped natural frequency ω_d, the subsidence (decrement) ratio δ, the damping ratio ζ, and finally ω_n = ω_d / √(1−ζ²). Stiffness was then calculated as K_s = J_lω_n².

Model validation — servo angle θ (Fig. 2)

Model validation — link angle α (Fig. 3)

State-space matrices A, B, C, D (Table 1)

Results & Analysis

From three successive peaks (O₁ = 20.48° at t = 0.244 s, O₂ = 13.38° at t = 0.592 s, O₃ = 9.70° at t = 0.924 s): T_osc = 0.34 s, ω_d = 18.48 rad/s, δ = 0.213, ζ = 0.0339.
Natural frequency: ω_n = 18.49 rad/s
Link stiffness: K_s = J_lω_n² = 0.0038 × 18.49² = 1.30 N·m/rad
Open-loop poles: P₁ = 0 (marginally stable); P₂,₃ = −8.1565 ± 22.5207i (stable oscillatory); P₄ = −24.0076 (stable).
The pole at zero indicates the system is marginally stable in open loop; the flexible link dynamics produce stable oscillatory modes at ≈22.5 rad/s.
Simulated and measured servo angle responses agreed well in rise time; the link angle showed correct frequency but slight amplitude deviation due to unmodeled damping and sensor noise.

Natural frequency calculation worksheet

Open-loop poles on s-plane

Transfer function expressions

MATLAB Code

Three scripts handled the free-oscillation plot, model validation comparison, and state-space / transfer function derivation. Full script: Lab 2 Appendix A.

% Experiment 1: Free-oscillation plot
figure();
plot(data_alpha(:,1), data_alpha(:,2), 'r', 'LineWidth', 1.5);
grid on;
xlabel('Time (s)');  ylabel('Angle (Deg)');
title('Free-Oscillation');

% Experiment 2: Model validation — servo angle
figure();
plot(data_theta(:,1), data_theta(:,2), 'LineWidth', 1.5);   % simulated
hold on
plot(data_theta(:,1), data_theta(:,3), 'LineWidth', 1.5);   % measured
xlabel('Time (s)');  ylabel('Servo Angle (Deg)');
title('Model Validation');
legend('Simulated Servo Angle', 'Measured')

% State-space → transfer function and poles
sys   = ss(A, B, C, D);
TF    = tf(sys)         % 2×1 transfer function [theta/u; alpha/u]
poles = eig(A)          % open-loop poles

Full MATLAB script — Lab 2 Appendix A

setup_flexgage.m setup script

q_flexgage_val Simulink model