M = 30;
d = 0.05;
h = 2;
w = 0.1;
b = 3;
L = 2;
Y = 30*10^9;
v = 0.3;
k = 3*Y*d^4/(12*L^3);
wn = Sqrt[k/M];
T = 2*Pi/wn;
deltaT = T/1000;
deltaT100 = deltaT*100;
Fhat = F*deltaT100;
ode = {M x2''[t] + k x2[t] == Fhat*DiracDelta[t - (1 - deltaT100/2)],
   x2[0] == 0, x2'[0] == 0};
solution = NDSolve[ode, x2[t], {t, 0, 100}];
hs[t] = (1/(M*wn))*Sin[wn*(t - (1 - deltaT100/2))]*
   HeavisideTheta[t - (1 - deltaT100/2)];
hs2[t] = (1/(M*wn))*Sin[wn*(t - (1 - deltaT/2))]*
   HeavisideTheta[t - (1 - deltaT/2)];

Plot[{x2[t] /. solution, hs[t], hs2[t]}, {t, 0, 2},
 PlotLabel ->
  "Convolution Response of Single DOF Undamped Oscillator",
 AxesLabel -> {"t", "x(t)"},
 PlotLegends -> {"Convolution",
   "Impulse Response \[DifferenceDelta]t = 100*\[DifferenceDelta]t*",
   "Impulse Response \[DifferenceDelta]t*"}]