• Derived equations of motion using Newton's and Lagrange’s methods

• Transformed high-order equations into state-space form (4 first-order ODEs)

• Computed natural frequencies (5.16 rad/s, 7.75 rad/s) and corresponding mode shapes

• Built free vibration simulations using MATLAB’s ode23 solver

• Applied modalfrf() to generate frequency response from randomized input

• Tuned damping and matrix inputs to reflect realistic system behavior

• MATLAB functions and derivations were completed from first principles and match theoretical solutions from class examples

• Code was submitted in-line per course requirement

• Modeled a 2-DOF system with masses, springs, and dampers (m₁, m₂, k₁–₃, c₁–₃)

• Represented both masses’ equations in matrix form: [M]{ẍ} + [C]{ẋ} + [K]{x} = {0}

• Linearized for modal and state-space analysis

%% Free Vibration Mode Calculation 

k1=2000; k2=2000; k3=5000;  

m1=100; m2=150;  

c1=2; c2=1.5; c3=0;  

m = [m1 0; 0 m2];  

k = [k1+k2 -k2; -k2 k2+k3];  

c = [c1+c2 -c2; -c2 c2+c3];  

[modeShape, fr] = eig(k, m); 

*Full code included in downloadable report

• Initial damping assumptions (e.g., c₃ = 0) caused underdamped responses; adjusted values for realistic decay

• Required conversion from second-order to first-order ODEs for ode23

• Manual matrix entry errors during FRF setup required debug iterations

• Tuning FFT resolution (window length, overlap) to match MATLAB’s modalfrf needs

• Derived equations of motion from first principles using Newtonian and Lagrangian approaches

• Linearized system for modal analysis and computed natural frequencies and eigenvectors

• Simulated damped motion of a 2-DOF system with validated natural frequencies and frequency response

• Evaluated frequency response function (FRF) using FFT-based random signal analysis with modalfrf

• Demonstrated full understanding of mechanical vibration analysis and digital simulation methods

MATLAB · System Modeling · Modal Analysis · Matrix Algebra · FRF · Damped Response · Eigenvalue Calculation · Simulation · State-Space Dynamics

This project strengthened my understanding of mechanical vibration principles by applying them through code. I practiced deriving equations of motion from first principles, translating systems into matrix form, and building simulations from scratch in MATLAB. I now have confidence in interpreting vibration modes and frequency behavior in second-order dynamic systems.