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This is my version of explanation. I would suggest you to come up with your own explanation. The best way would be for you to try explain this to somebody else in your own words.

Following is my version of explanation, but this is just an example. You may come up with a better version.

Resonance - Spring Mass

Click on Next and Prev button so that some intuitive images forms in your head.

< Code 1 >

m = 1.0;

k = 0.75;

c = 0.7;

F0 = 0.5;

tmax = 100;

yinit = 0.0;

hFig = figure(1,'Position',[300 300 700 600]);  

n = 0;

w = 0.05 * n;

wn = sqrt(k/m);

zeta = c/(2*sqrt(k*m));

dy_dt = @(t,y) [y(2);...

                 -(c/m) * y(2) - (k/m) * y(1) + (F0/m) * sin(w*t)];

odeopt = odeset ('RelTol', 0.00001, 'AbsTol', 0.00001,'InitialStep',0.5,'MaxStep',0.5);

[t,y] = ode45(dy_dt,[0 tmax], [yinit 0.0],odeopt);

subplot(3,7,[1 7] );

tx = 0.2;

text(tx,1.2,"m x''(t) + c x'(t) + k x(t) = Fo sin(\\omega t)",'FontSize',16,'fontweight','bold');

tx = 0.2;

tStr = sprintf("m = %0.2f, k = %0.2f, c = %0.2f, Fo = %0.2f",m,k,c,F0);

text(tx,0.9,tStr,'FontSize',14);

tStr = sprintf("%0.2f x''(t) + %0.2f x'(t) + %0.2f x(t) = %0.2f sin(%0.2f t)", ...

                m,c,k,F0,w);

text(tx,0.6,tStr,'FontSize',14);

tx = 0.5;

tStr = sprintf("\\omega_n = \\surd{(k/m)} = \\surd{(%0.2f/%0.2f)} = %0.2f",k,m,wn);

text(tx,0.2,tStr,'FontSize',14);

tStr = sprintf("\\zeta = c/[2 \\surd(k m)] = %0.2f/[2 \\surd(%0.2f %0.2f)] = %0.2f",c,k,m,zeta) ;

text(tx,-0.1,tStr,'FontSize',14);

wRatio = w/wn;

tStr = sprintf("\\omega / \\omega_n = %0.2f / %0.2f = %0.2f",w,wn,wRatio) ;

text(tx,-0.4,tStr,'FontSize',14);

axis([0 7 0 1]);

set(gca,'Visible','off')

subplot(3,7,[8 12] );

%plot(t,y(:,1),'r-',t,y(:,2),'b-',t,sin(w*t),'k--');

plot(t,y(:,1),'r-',t,y(:,2),'b-');

ylim([-3 3]);

legend('y(1)','y(2)');

xlabel('time');

ylabel('Amplitude');

set(gca,'xticklabel',[]);

set(gca,'yticklabel',[]);

subplot(3,7,[13 14]);

plot(y(:,1),y(:,2),'b-');

xlim([-3 3]); ylim([-3 3]);

xlabel('y(1)');

ylabel('y(2)');

set(gca,'xticklabel',[]);

set(gca,'yticklabel',[]);

subplot(3,7,[15 21]);

w = 0:0.01:5;

nw = w ./ wn;

Amp = (F0/k) ./ sqrt((1 - (nw .^ 2)).^2 + (2 * zeta * nw).^2 );

Ap = (F0/k) ./ sqrt((1 - (wRatio .^ 2)).^2 + (2 * zeta * wRatio).^2 );

hold on;

plot(nw,Amp,'r-');

plot([wRatio wRatio],[0 Ap],'k--');

plot([wRatio],[Ap],'bo','MarkerFaceColor',[0 0 1],'MarkerSize',8);

xlabel('\omega / \omega_n');

ylabel('Steady State Amplitude');

axis([0 3 0 2]);

set(gca,'yticklabel',[]);

grid on;

hold off;