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 . If we assume that each spring obeys Hooke's law, Newtons law gives us the following set of differential equations:

+

+

-

In matrix form, this system can be written:

M x ''(t) = K x (t)

where the mass matrix

M =

and the stiffness matrix

K =

x ''(t) = A x (t)

where = M K .

If the 3 x 3 matrix A has distinct negative eigenvalues with associated real eigenvectors V1, V2 and V3 , then the solution of x ''(t) = A x (t) is given by

x (t) = V i

• SpringMassCouplet (L, init ) solves the second order system described above, displays the matrix = M K A, the eigenvalues and the corresponding circular frequencies. The motion of the spring-couplet masses can be animated in four different cases:
  1) Three masses and four springs: L = [ , , , ]
  2) Three masses and three springs, no spring connected to the right-hand wall: L = [ , , ]
  3) Two masses and three springs: L = [ , , ]
  4) Two masses and two springs, no spring connected to the right-hand wall: L = [ , ]
  The initial conditions is defined by init = [ ]

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.