14.3: Normal Modes

The normal modes of the two-coupled oscillator system are obtained by a transformation to a pair of normal coordinates \((\eta_1, \eta_2)\) that are independent and correspond to the two normal modes. The pair of normal coordinates for this case are

\[\begin{align} \eta_1 \equiv x_1 − x_2 \label{14.12}\\ \eta_2 \equiv x_1 + x_2 \notag \end{align}\]

that is

\[\begin{align} x_1 = \frac{1}{2} (\eta_2 + \eta_1) \label{14.13}\\ x_2 = \frac{1}{2} (\eta_2 − \eta_1) \notag \end{align}\]

Substitute these into the equations of motion \((14.2.1)\), gives

\[\begin{align} m (\ddot{\eta}_1 + \ddot{\eta}_2 ) + (\kappa + 2\kappa^{\prime} ) \eta_1 + \kappa^{\prime} \eta_2 = 0 \\ \notag m (\ddot{\eta}_1 − \ddot{\eta}_2 ) + (\kappa + 2\kappa^{\prime} ) \eta_1 − \kappa^{\prime} \eta_2 = 0 \end{align}\]

Adding and subtracting these two equations gives

\[\begin{align} m\ddot{\eta}_1 + (\kappa + 2\kappa^{\prime} ) \eta_1 = 0 \\\notag m\ddot{\eta}_2 + \kappa \eta_2 = 0 \end{align}\]

Note that the two coordinates \(\eta_1\) and \(\eta_2\) are uncoupled and therefore are independent. The solutions of these equations are

\[\begin{align} \eta_1 (t) = C^+_1 e^{i\omega_1 t} + C^−_1 e^{-i\omega_1 t} \\ \eta_2 (t) = C^+_2 e^{i\omega_2 t} + C^−_2 e^{-i\omega_2 t} \end{align}\]

where \(\eta_1\) corresponds to angular frequencies \(\omega_1\), and \(\eta_2\) corresponds to \(\omega_2\). The two coordinates \(\eta_1\) and \(\eta_2\) are called the normal coordinates and the two solutions are the normal modes with corresponding angular frequencies, \(\omega_1\) and \(\omega_2\).

Figure \(\PageIndex{1}\): Motion of two coupled harmonic oscillators in the \((x_1, x_2)\) spatial configuration space and in terms of the normal modes \((\eta_1, \eta_2)\). Initial conditions are \(x_2 = D, x_1 = \dot{x}_1 = \dot{x}_2 = 0\).

The \((\eta_1, \eta_2)\) axes of the two normal modes correspond to a rotation of \(45^{\circ}\) in configuration space, Figure \(\PageIndex{1}\). The initial conditions chosen correspond to \(\eta_1 = −\eta_2\) and thus both modes are excited with equal intensity. Note that there are 5 lobes along the \(\eta_2\) axis versus 4 lobes along the \(\eta_1\) axis reflecting the ratio of the eigenfrequencies \(\omega_1\) and \(\omega_2\). Also note that the diamond shape of the motion in the \((x_1, x_2)\) configuration space illustrates that the extrema amplitudes for \(x_2\) are a maximum when \(x_1\) is zero, and vise versa. This is equivalent to the statement that the energies in the two modes are coupled with the energy for the first oscillator being a maximum when the energy is a minimum for the second oscillator, and vise versa. By contrast, in the \((\eta_1, \eta_2)\) configuration space, the motion is bounded by a rectangle parallel to the \((\eta_1, \eta_2)\) axes reflecting the fact that the extrema amplitudes, and corresponding energies, for the \(\eta_1\) normal mode are constant and independent of the motion for the \(\eta_2\) normal mode, and vise versa. The decoupling of the two normal modes is best illustrated by considering the case when only one of these two normal modes is excited. For the initial conditions \(x_1 (0) = −x_2 (0)\), and \(\dot{x}_1 (0) = − \dot{x}_2 (0)\), then \(\eta_2 (t)=0\). That is, only the \(\eta_1 (t)\) normal mode is excited with frequency \(\omega_1\) which corresponds to motion confined to the \(\eta_1\) axis of Figure \(\PageIndex{1}\).

Figure \(\PageIndex{2}\): Normal modes for two coupled oscillators.

As shown in Figure \(\PageIndex{2}\), \(\eta_1 (t)\) is the antisymmetric mode in which the two masses oscillate out of phase such as to keep the center of mass of the two masses stationary. For the initial conditions \(x_1 (0) = x_2 (0)\), and \(\dot{x}_1 (0) = \dot{x}_2 (0)\), then \(\eta_1 (t)=0\), that is, only the \(\eta_2 (t)\) normal mode is excited. The \(\eta_2 (t)\) normal mode is the symmetric mode where the two masses oscillate in phase with frequency \(\omega_2\); it corresponds to motion along the \(\eta_2\) axis. For the symmetric phase, both masses move together leading to a constant extension of the coupling spring. As a result the frequency \(\omega_2\) of the symmetric mode \(\eta_2 (t)\) is lower than the frequency \(\omega_1\) of the asymmetric mode \(\eta_1 (t)\). That is, the asymmetric mode is stiffer since all three springs provide active restoring forces, compared to the symmetric mode where the coupling spring is uncompressed. In general, for attractive forces the lowest frequency always occurs for the mode with the highest symmetry