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% Exploring natural and resonant frequencies of harmonic oscillators.
% Samped, driven harmonic oscillators
% ref: https://en.wikipedia.org/wiki/Harmonic_oscillator
% parameters
f0 = 10/1e3; % undamped "natural" frequency
m = 1; % mass
l = 0; % damping ratio
w0 = 2*pi*f0; % rad/ms, undamped angular frequency
F0 = w0^2; % amplitude of driving force
% step input
eqns_harmonic_step={
'du/dt=v; u(0)=.5'
'dv/dt=-w0^2*u-2*l*w0*v+(1/m)*F0'
};
% sinusoidal driving force
eqns_harmonic_sine={
'du/dt=v; u(0)=.5'
'dv/dt=-w0^2*u-2*l*w0*v+(1/m)*F0*sin(2*pi*f*t)'
};
% step response simulations
vary={'m',m;'l',l;'w0',w0;'F0',F0};
data=dsSimulate(eqns_harmonic_step,'vary',vary,'time_limits',[0 1000]);
dsPlot(data);
% vary input strength
vary={'m',m;'l',l;'w0',w0;'F0',F0*[1 2 3]};
data=dsSimulate(eqns_harmonic_step,'vary',vary,'time_limits',[0 1000]);
dsPlot(data);
% frequency response simulations
vary={'m',m;'l',l;'w0',w0;'F0',F0;'f',[0:2:20]/1e3};
data=dsSimulate(eqns_harmonic_sine,'vary',vary,'time_limits',[0 1000]);
dsPlot(data);
% vary input strength
vary={'m',m;'l',l;'w0',w0;'F0',F0*[1 2 3];'f',[0:2:20]/1e3};
data=dsSimulate(eqns_harmonic_sine,'vary',vary,'time_limits',[0 1000]);
dsPlot(data);
% observation 1:
% - resonant frequency = undamped frequency given equal strength step input
% TRUE for all second-order linear oscillatory systems.
% observation 2:
% - resonant frequency is independent of input strength
% - response amplitude increases with input strength