How to combine more plots in single image inside a loop

Question

I was reading basic help here How to | Combine Two or More Graphics , but I did not get it.

I have simple differential equation and want to change a constant "b" (in this case "damping") and plot all graphs in a single image. Here is the code:

(*Single mass damper system of a rail car *)
(*with 10 mm starting deflection *)
m = 40000;     (* [kg]    *)
k = 1600000; (* [N/m]   *)

for[b=0, b<=160000, b=b+20000,

sol = NDSolve[{
m*x''[t] + b*x'[t] + k*x[t]== 0,
x'[0] == 0,x[0] == 0.01},
x[t], {t, 0, 10}];

Print[Plot[x[t] /. sol, {t, 0, 3}, 
PlotRange -> Full ,
PlotTheme -> {"Grid"},
AxesLabel->{"Time [s]", "Vertical suspension deflection [m]"}, 
PlotLabel->Style["Single mass damper system of a rail car", 12, Bold]]] 
]

Can anyone please help? I'm just missing "hold on" from Matlab here. Thank you in advance.

Answer 1

Clear["Global`*"];

m = 40000;
k = 1600000;

Your equation can be solved exactly with DSolve

sol = DSolve[{m*x''[t] + b*x'[t] + k*x[t] == 0, x'[0] == 0, x[0] == 1/100}, 
    x[t], t][[1]] // Simplify

Alternatively,

sol2 = sol // ExpToTrig // Simplify

Plotting,

Plot[Evaluate@Table[x[t] /. sol,
   {b, 0, 160000, 20000}], {t, 0, 3},
 PlotTheme -> "Grid",
 Frame -> True,
 FrameLabel -> (Style[#, 12, Bold] & /@
    {"Time [s]", 
     "Vertical suspension deflection [m]"}),
 PlotLabel -> 
  Style["Single mass damper system of a rail car", 12, Bold],
 PlotLegends ->
  LineLegend[Range[0, 160000, 20000],
   LegendLabel -> Style["b =", 12, Bold]]]