Clear[A,m,b,k,w,x,X]
DSolve[{A Sin[w t] == m x''[t]+b x'[t]+k x[t],x[0]==0,x'[0]==0},x[t],t];
x[t]/.%;
%[[1]];
Re[%];
X[t]=FullSimplify[%,Assumptions->{A≠0,m>0,m≠0,b>0,b≠0,k>0,k≠0,b^2-4k m<0,b^2-4k m≠0}]

Clear[A,m,b,k,w]

A=1;
m=1;
b=.01;
k=100;
w=20/(Pi);

FullSimplify[X[t]]

Plot[%,{t,0,10}]