Consider the spring-mass system, shown in Figure \(4.2 .4,\) consisting of two
unit masses suspended from springs with spring constants 3 and \(2,\)
respectively. Assume that there is no damping in the system.
(a) Show that the displacements \(u_{1}\) and \(u_{2}\) of the masses from their
respective equilibrium positions satisfy the equations
$$
u_{1}^{\prime \prime}+5 u_{1}=2 u_{2}, \quad u_{2}^{\prime \prime}+2 u_{2}=2
u_{1}
$$
(b) Solve the first of Eqs. (i) for \(u_{2}\) and substitute into the second
equation, thereby obtaining the following fourth order equation for \(u_{1}:\)
$$
u_{1}^{\mathrm{iv}}+7 u_{1}^{\prime \prime}+6 u_{1}=0
$$
Find the general solution of Eq. (ii).
(c) Suppose that the initial conditions are
$$
u_{1}(0)=1, \quad u_{1}^{\prime}(0)=0, \quad u_{2}(0)=2, \quad
u_{2}^{\prime}(0)=0
$$
Use the first of Eqs. (i) and the initial conditions (iii) to obtain values
for \(u_{1}^{\prime \prime}(0)\) and \(u_{1}^{\prime \prime \prime}(0)\) Then show
that the solution of Eq. (ii) that satisfies the four initial conditions on
\(u_{1}\) is \(u_{1}(t)=\cos t .\) Show that the corresponding solution \(u_{2}\) is
\(u_{2}(t)=2 \cos t .\)
(d) Now suppose that the initial conditions are
$$
u_{1}(0)=-2, \quad u_{1}^{\prime}(0)=0, \quad u_{2}(0)=1, \quad
u_{2}^{\prime}(0)=0
$$
Proceed as in part (c) to show that the corresponding solutions are
\(u_{1}(t)=-2 \cos \sqrt{6} t\) and \(u_{2}(t)=\cos \sqrt{6} t\)
(e) Observe that the solutions obtained in parts (c) and (d) describe two
distinct modes of vibration. In the first, the frequency of the motion is \(1,\)
and the two masses move in phase, both moving up or down together. The second
motion has frequency \(\sqrt{6}\), and the masses
move out of phase with each other, one moving down while the other is moving
up and vice versa. For other initial conditions, the motion of the masses is a
combination of these two modes.
