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Two-Story-Building-Free-Vibration-Simulation

This MATLAB code simulates the free vibration of a two-story building modeled with coupled second-order differential equations. It uses Euler’s Method (EM), Improved Euler’s Method (IEM), and an analytical solution to compute and compare the displacements of the building’s floors over time.

Problem Statement

The behavior of the two-story building is governed by the following coupled second-order differential equations:

m1 x1'' = -k1 x1 - k2 (x1 - x2)

m2 x2'' = -k2 (x2 - x1)

Parameters

The physical parameters of the system are:

  • Spring stiffness: ( k1 = k2 = 4.66 \ kN/mm)
  • Mass of the first floor: ( m1 = 0.0917 )
  • Mass of the second floor: ( m2 = 0.0765 )

Initial Conditions

  • x1(0) = 100 , x2(0) = 50
  • x1'(0) = 0, x2'(0) = 0

Goal

Simulate the behavior of the system over the interval ( t in [0, 10] ) using the following approaches:

  1. Analytical solution for the given parameters.
  2. Numerical solutions using:
    • Euler's Method (EM)
    • Improved Euler's Method (IEM)

MATLAB Implementation

Features

  1. Numerical Methods:
    • Euler's Method (EM)
    • Improved Euler's Method (IEM)
  2. Comparison with Analytical Solution.
  3. Plots:
    • Solutions from EM, IEM, and analytical methods.
    • Different numbers of time steps (( N = 45, 55, 75, 100, 250 )).

MATLAB Code

The repository includes:

  1. EMsolver: Solves the system using Euler's Method.
  2. IEMsolver: Solves the system using Improved Euler's Method.
  3. analytical: Computes the analytical solution.

How to Use

  1. Clone the repository:

    git clone https://github.com/yourusername/two-story-vibration.git

  2. Open the MATLAB script in the repository.

  3. Execute the script to:

    • Compute the solutions using EM, IEM, and analytical approaches.

    • Plot the results for different time step values.