Computer Algebra Recipes

140 CHAPTER 3. LINEAR ODE MODELS
The boundary condition implies that there are certain allowed frequencies (eigenfrequencies)
! of vibration, each frequency corresponding to a possible normal
mode solution. A general transverse motion of the cord will involve a linear combination
of normal modes, the combination depending on the initial conditions
imposedonthecord.
The eigenfrequencies will now be determined. First, the op command is
used to extract the ¯rst element of the second argument on the lhs of bc.
> c:=op([2,1],lhs(bc));
c := 3:499271060
Using the BesselJZeros command, the ¯rst 10 eigenfrequencies are numerically
evaluated, and assigned.
> freq:=seq(omega[n]=evalf(BesselJZeros(0,n)/c),n=1..10);
freq := !1 =0:6872361462; !2 =1:577493717; !3 =2:473008739;
!4 =3:369711645; !5 =4:266865142; !6 =5:164236682;
!7 =6:061730076; !8 =6:959298411; !9 =7:856916100;
!10 =8:754568007
> assign(freq):
A functional operator for creating the nth normal mode, with the time dependence
included, is formed.
> mode:=n->eval(X,omega=omega[n])*cos(omega[n]*t):
As an explicit example, choosing n = N = 3 produces the normal mode X3
corresponding to the third eigenfrequency.
> N:=3: X[N]:=mode(N);
X3 := BesselJ(0; 4:946017478 p 3:061224489 ¡ :1020408163 y)cos(2:473008739 t)
The normal mode is now animated, the plot being rotated by ¡ ¼
2 radians using
the rotate command, so that the animated string is hanging vertically in equilibrium,
rather than being pictured as horizontal. The scaling is constrained,
so that the transverse displacement is small compared to the length of the cord.
The axes are also removed so that the small vibrations are better viewed.
> rotate(animate(plot,[X[N],y=0..L],t=0..10,color=red,
frames=100,scaling=constrained,axes=none),-Pi/2);
The animation can be observed on your computer by executing the above command
line, clicking on the computer plot, and on the start arrow in the tool
bar. You should also look at the other transverse vibrational modes as well, by
changing the value of N from 3 to some other integer between 1 and 10. Or, if
you want, you could generate even higher-integer normal modes.
While we have been looking at Jennifer's computer algebra treatment of the
transverse vibrations of the cord, Heather has completed her bungee jump, the
largest of the vertical oscillations bringing her head to within a meter of the
river far below the bridge. Looking rather pale, Heather attempts to persuade
Jennifer to also do a bungee jump, but Jennifer sensibly declines.