Normal modes

The technique to solve such coupled differential equations is to identify linear combinations of $x_0$ and $x_1$ for which the equations become decoupled. In this case it is very easy to identify such variables

(5.5)

These are referred to as as the normal modes (or eigen modes) of the system and the equations governing them are

$$m \frac{d^2 \, q_0}{dt^2} = - k q_0$$ (5.6)

and

$$m \frac{d^2 \, q_1}{dt^2} = - (k + 2 k^{'}) q_1 \,.$$ (5.7)

The two normal modes execute simple harmonic oscillations with respective angular frequencies

$$\omega_0=\sqrt{\frac{k}{m}} \, \, \hspace{0.5cm} {\rm and} \,\, \hspace{0.5cm} \omega_1=\sqrt{\frac{k+2k'}{m}}$$ (5.8)

In this case the normal modes lend themselves to a simple physical interpretation where.

The normal mode $q_0$ represents the center of mass. The center of mass behaves as if it were a particle of mass $2 m$ attached to two springs (Figure 5.2) and its oscillation frequency is the same as that of the individual decoupled oscillators $\omega_0=\sqrt{\frac{2 k}{2m}}$.

Figure 5.2: This shows the spring mass equivalent of the normal mode $q_0$ which corresponds to the center of mass.

The normal mode $q_1$ represents the relative motion of he two masses which leaves the center of mass unchanged. This can be thought of as the motion of two particles of mass $m$ connected to a spring of spring constant $\tilde{k}=(k+2 k^{'})/2$ as shown in Figure 5.3. The oscillation frequency of this normal mode $\omega_1=\sqrt{\frac{k+2 k^{'}}{m}}$ is always higher than that of the individual uncoupled oscillators (or the center of mass). The modes $q_0$ and $q_1$ are often referred to as the slow mode and the fast mode respectively.

Figure 5.3: This shows the spring mass equivalent of the normal mode $q_1$ which corresponds to two particles connected through a spring.

We may interpret $q_0$ as a mode of oscillation where the two masses oscillate with exactly the same phase, and $q_0$ as a mode where they have a phase difference of $\pi$ (Figure 5.4). Recollect that the phases of the two masses are independent when the two masses are not coupled. Introducing a coupling causes the phases to be interdependent.

Figure 5.4: This shows the motion corresponding to the two normal modes $q_0$ and $q_1$ respectively.

The normal modes have solutions

(5.9)

$$\tilde{q}_1(t) = \tilde{A}_1 \ e^{i \ \omega_1 t}$$ (5.10)

where it should be bourne in mind that $\tilde{A}_0$ and $\tilde{A}_1$ are complex numbers with both amplitude and phase ie.     $\tilde{A}_0 = A_0 e^{i \psi_0}$ etc. We then have the solutions

$$\tilde{x}_0(t)= \tilde{A}_0 \ e^{i \ \omega_0 t} + \tilde{A}_1 \ e^{i \ \omega_1 t}$$ (5.11)

$$\tilde{x}_1(t) =\tilde{A}_0 \ e^{i \ \omega_0 t} - \tilde{A}_1 e^{i \ \omega_1 t}$$ (5.12)

The complex amplitudes $\tilde{A}_1$ and $\tilde{A}_2$ have to be determined from the initial conditions, four initial conditions are required in total.