Computer Algebra Recipes

3.2. SECOND-ORDER MODELS 123
Loading the plots package, we use the alias command. Entering x1, x2, m1,
m2 will produce the subscripted quantities x1, x2, m1, andm2in the output.
> restart: with(plots):
> alias(x[1]=x1,x[2]=x2,m[1]=m1,m[2]=m2):
When m2 is displaced from equilibrium by amount x2 at time t in, say, the
positive x-direction, it exerts a force on m1 through the connecting spring. If
m1 is displaced from equilibrium by amount x1(t), the force exerted on m1 by
m2 is k (x2(t) ¡ x1(t)). The spring connected to the wall will exert a force
¡kx1(t) in the opposite direction. Including damping, Newton's second law
yields the ODE ode1 governing the displacement x1(t) ofm1.
> ode1:=m1*diff(x1(t),t,t)
=k*(x2(t)-x1(t))-k*x1(t)-a*diff(x1(t),t);
μ
μ 2 d d
ode1 := m1 x1(t) = k (x2(t) ¡ x1(t)) ¡ k x1(t) ¡ a
dt2 dt x1(t)
The mass m1 will exert an equal and opposite force on m2 through the connecting
spring, i.e., ¡k (x2(t) ¡ x1(t)). Including the damping and driving forces,
the displacement x2(t) ofm2 is given by the ODE ode2 .
> ode2:=m2*diff(x2(t),t,t)
=-k*(x2(t)-x1(t))-a*diff(x2(t),t)+f*sin(omega*t);
μ
μ 2 d d
ode2 := m2 x2(t) = ¡k (x2(t) ¡ x1(t)) ¡ a
dt2 dt x2(t)
+ f sin(!t)
One has a system of two coupled, linear, second-order, inhomogeneous ODEs.
To solve them, let's ¯rst enter the given parameter values,
> m1:=2: m2:=1: k:=1: a:=1: omega:=2: f:=2:
and the initial conditions.
> ic:=x1(0)=1,x2(0)=0,D(x1)(0)=0,D(x2)(0)=0:
The set of ODEs is analytically solved for x1(t) andx2(t), subject to the initial
conditions, using the dsolve command with the Laplace transform option.
The same answer could be obtained by mimicking the hand calculation in the
previous example. This is left as a problem.
> sol:=dsolve(fode1,ode2,icg,fx1(t),x2(t)g,method=laplace);
sol := fx1(t) = 36
26
cos(2 t)+ sin(2 t)
493 493
¡ 1 X
( (56380 + 339955 ® + 193087 ®
194242
®=%1
2 + 188250 ® 3 ) e ( ®t) )g;
fx2(t) =¡ 164 228
cos(2 t) ¡ sin(2 t)
493 493
¡ 1 X
( (107923 + 533529 ® + 249954 ®
194242
®=%1
2 + 293736 ® 3 ) e ( ®t) )g
%1 := RootOf(2 Z 4 +3 Z 3 +5 Z 2 +3 Z +1)