Underdamped Motion

We select = 1 and two r values ( ).

> plt3:=plot([xu(1,1/2,1,2,t),xu(1,1/5,1,2,t)],t=0..20):

> txt3:=textplot([ [6.5,xu(1,1/2,1,2,6.5),`r=1/2`],[6.5,xu(1,1/5,1,2,6.5),`r=1/5`]]):

> pltod:=display(plt3,txt3,labels=[`t`,`x(t)`],labelfont=[TIMES,BOLD,12]):%;

Figure 7 Underdamped motion, m = 1 kg, k = 1 , , r: damping constant

> mass_spring(1,1/5,1,0,5,2,0,20,0,60,scaling=constrained);

Figure 8 Animation of an underdamped motion of a mass-spring system with dashpot. m = 1 kg, k = 1 , r =

The action of the dashpot exponentially damps the oscillations in accord with with the time-varying amplitude. The dashpot also decreases the frequency of the motion from 1 in the undamped to case in the underdamped motion with the same mass and force constant.

Comparison of the graphs in Figure 2, 5 and 7 (Figure 9) shows that when the motion is critically damped the mass reaches its equilibrium posistion in a shorter time than when the damping is larger. And for any damping constant less than in our particular example the motion becomes oscillatory.

> display(plt1(45),plt2,plt3,txt1(45),txt2,txt3,labels=[`t`,`x(t)`],labelfont=[TIMES,BOLD,12]);

Figure 9 Overdamped, critically damped and damped oscillatary motion of a mass-spring system.
m = 1 kg, k = 1 , , r: damping constant