Honors Physics 1 Class 20 Fall 2013 Coupled Oscillators Normal Modes Fourier Analysis

Damped Harmonic Oscillator

Two masses, three springs, tethered at both ends

Coupled oscillators – general solution

Coupled Oscillators - cases

Coupled oscillators – Case 1

Coupled oscillators – Case 1

Coupled oscillators – Case 1 Three equal springs

Fourier analysis We introduce this powerful tool as a technique for determining from experimental data the normal mode oscillation frequencies.

A MATLAB program for Fourier analysis % This is an example of the use of the Fourier transform for finding components of a signal. clear all %Get the data and put it in arrays %The Excel file must be in the same directory as the MATLAB program A=xlsread('fourier_transform_example_data'); %Reads an array of data from an excel file of this name t = A(:,1); %Put the time data from the first column into a vector y = A(:,4); %Put the position data in the fourth column into a second matched vector Fs=1/(t(2)-t(1)); %Set the sampling frequency to 1/time step L = length(t); % Length of signal vector %Plot raw data h=figure(1); % Set up the figure axes('FontSize',16,'FontName','Times New Roman','LineWidth',.5,'Box','on','xLim',[0 20], 'yLim',[0 10]) line(t,y,'LineStyle','-','LineWidth',2) %Plot the data xlabel('time'); ylabel('position') % The Fourier transform part of the program NFFT = 2^nextpow2(L); % Next power of 2 from length of y Y = fft(y,NFFT)/L; f = Fs/2*linspace(0,1,NFFT/2+1); %Plot transform h=figure(2); line(f,2*abs(Y(1:NFFT/2+1)),'LineStyle','-','LineWidth',2) xlabel('frequency'); ylabel('amplitude')