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, [Maple Math] and [Maple Math] connected to each other and to two walls by four springs with spring constants [Maple Math] and [Maple Math] .

[Maple Plot]

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 [Maple Math] and [Maple Math] of the respective masses from their equilibrium posistions as coordinates. The first spring is then stretched the distance [Maple Math] , the second spring is stretched the distance [Maple Math] , the third spring is stretched the distance [Maple Math] and the fourth spring is stretched the distance [Maple Math] . 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);

[Maple Math]
[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]
[Maple Math]
[Maple Math]

[Maple Plot]

Figure 5 Three spring-couplet masses.
The vertical lines marks the equilibrium positions of the masses.
[Maple Math] = [Maple Math] = [Maple Math] = 1 kg , [Maple Math] [Maple Math] , [Maple Math] = [Maple Math] [Maple Math] and [Maple Math] [Maple Math] .

Figure 5 shows that the natural frequencies of the system is [Maple Math] .In the first natural mode the two masses [Maple Math] and [Maple Math]move in opposite directions with equal amplitudes. The mass [Maple Math]move in the same direction but with the amplitude of motion half that of [Maple Math]. In the second mode [Maple Math]and [Maple Math]move in the same directions, opposite to [Maple Math], with the amplitude [Maple Math]twice that of [Maple Math]and equal to that of [Maple Math]. In the last mode with frequency [Maple Math]all three masses move in the same direction.[Maple Math] and [Maple Math]have equal amplitudes twice that of [Maple Math] .

Two masses and three springs

> L:=[[1,1],[1,4],1]:init:=[1,2,0,0]:

> SpringMassCouplet(L,init,16,scaling=unconstrained,axes=frame);

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Plot]

Figure 6 Three spring-couplet masses .The vertical lines
marks the equilibrium positions of the masses.
[Maple Math] = 1 kg , [Maple Math] = 1 kg, [Maple Math][Maple Math] , [Maple Math] [Maple Math] and [Maple Math][Maple Math]

Figure 6 shows that the natural frequencies of the system is [Maple Math] and [Maple Math] . 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.