Forced Multi-Mode Oscillator System on MATLAB
Skills: MATLAB simulations, Lagrangian Equations
This project aims to study the behavior of materials under cyclic loads by simulating is the oscillation of a 4 point-masses system with dampers. This study was inspired by the Millennium Bridge, which wobbled due to the rhythmic marching of footsteps, and it had to be closed. I worked in a group of three on this project.
Technical Description
The system has 4 masses connected to each other and the wall by 5 springs and 5 dampers as shown below. The 4 masses are assumed to point masses with masses mi, connected by springs ki and ki+1 and unstretched length L0, and connected by dampers bi and bi+1. The first mass is acted on by a cyclic force F1(𝑡).
Mathematical Model
It is assumed that the springs and dampers are massless, while the masses are point masses which do not experience any rotation during the movement. Also, the masses only move vertically in the y direction and that there is no movement in the x direction. Lastly, L0, free length of spring is taken as 0. The Lagrangian approach was used to derive the system’s equations of motion. This system has 4 degrees of freedom, with generalized coordinates y1, y2, y3 and y4 denoting the y position of the 4 masses. y0, ẏ0, y5, ẏ5 = 0 in the equations as they are fixed connection points of the spring and dampers mounted on the wall.
The general equation for the respective masses can be presented in the following form.
Results
MATLAB was used to simulate the above scenario. As expected, the system experienced resonance its four natural frequencies when the damping constants were low.
Varying Damping Constants
Varying Spring Constant