Spring-Coupled Masses
The purpose of this section is to describe and
animate the motion of two or three masses connected to attached
springs. Figure 4 shows three masses, and connected to each
other and to two walls by four springs with spring constants and .
Figure 4 Three
spring-couplet masses
Since friction is to be neglected, the only
forces acting on the masses are those due to the extension and
compression of the attached springs. We take the rightward
displacements and of the respective masses from their equilibrium
posistions as coordinates. The first spring is then stretched the
distance ,
the second spring is stretched the distance , the third spring
is stretched the distance and the fourth spring is stretched the distance . We assume that
each spring obeys Hooke's law.
Three masses and four springs
> L:=[[1,1],[1,2],[1,2],3]:init:=[-1.5,1,0,0,0,0]:
> SpringMassCouplet(L,init,10,scaling=unconstrained,axes=frame);
Figure 5 Three
spring-couplet masses.
The vertical lines marks the equilibrium positions of the masses.
=
= = 1 kg , , = and .
Figure 5 shows that
the natural frequencies of the system is .In the first
natural mode the two masses and move in opposite directions with equal amplitudes.
The mass move
in the same direction but with the amplitude of motion half that
of . In
the second mode and move in the same directions, opposite to , with the amplitude
twice
that of and
equal to that of . In the last mode with frequency all three masses
move in the same direction. and have equal amplitudes twice that of .
Two masses and three springs
> L:=[[1,1],[1,4],1]:init:=[1,2,0,0]:
> SpringMassCouplet(L,init,16,scaling=unconstrained,axes=frame);
Figure 6 Three
spring-couplet masses .The vertical lines
marks the equilibrium positions of the masses.
=
1 kg , = 1 kg, , and
Figure 6 shows that
the natural frequencies of the system is and . In the first
natural mode the two masses move in opposite directions with
equal amplitudes. In the second they move in the same direction
with equal amplitudes of oscillation.