Example 3.7.6

Check the solution of the initial value problem

k=19600
m=100
sol=DSolve[{m*u''[t]+k*u[t]==0,u[0]==0,u'[0]==10},u[t],t]

Plot[(5/7) Sin[14 t],{t,0,Pi/2},GridLines->{{0,Pi/14,Pi/7},None}]

Example 3.7.11

Calculating the constants of intergation:

v[t_]=c1*Exp[-3t/20] Cos[(Sqrt[5991])t/20]+c2*Exp[-3t/20] Sin[(Sqrt[5991])t/20]

Solve[{v[0]==1/20,v'[0]==1/10},{c1,c2}]

Check the solution of the initial value problem

k=30
m=2
gam=3/5
sol=DSolve[{m*u''[t]+gam*u'[t]+k*u[t]==0,u[0]==1/20,u'[0]==1/10},u[t],t]

Quasifrequency:

(Sqrt[1-(((3/5)^2)/(4(30)(2)))])((Sqrt[5991])/20)

Simplify[%]

N[%]

Plot[(((E^(-3 t/20)) (5991 Cos[(((Sqrt[5991]) t)/20)]+43 (Sqrt[5991]) Sin[(((Sqrt[5991]) t)/20)]))/119820),{t,0,6}]

Let's solve the undamped case using Mathematica:

k=30
m=2
gam=0
sol=DSolve[{m*u''[t]+gam*u'[t]+k*u[t]==0,u[0]==1/20,u'[0]==1/10},u[t],t]

Plot that shows the effects of damping on the solution:

Plot[{(((E^(-3 t/20)) (5991 Cos[(((Sqrt[5991]) t)/20)]+43 (Sqrt[5991]) Sin[(((Sqrt[5991]) t)/20)]))/119820),(1/300) (15 Cos[(Sqrt[15]) t]+2 (Sqrt[15]) Sin[(Sqrt[15]) t])},{t,0,6},PlotStyle->{Red, Blue}]