14.3: Normal Modes
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The normal modes of the two-coupled oscillator system are obtained by a transformation to a pair of normal coordinates (η1,η2) that are independent and correspond to the two normal modes. The pair of normal coordinates for this case are
η1≡x1−x2η2≡x1+x2
that is
x1=12(η2+η1)x2=12(η2−η1)
Substitute these into the equations of motion (14.2.1), gives
m(¨η1+¨η2)+(κ+2κ′)η1+κ′η2=0m(¨η1−¨η2)+(κ+2κ′)η1−κ′η2=0
Adding and subtracting these two equations gives
m¨η1+(κ+2κ′)η1=0m¨η2+κη2=0
Note that the two coordinates η1 and η2 are uncoupled and therefore are independent. The solutions of these equations are
η1(t)=C+1eiω1t+C−1e−iω1tη2(t)=C+2eiω2t+C−2e−iω2t
where η1 corresponds to angular frequencies ω1, and η2 corresponds to ω2. The two coordinates η1 and η2 are called the normal coordinates and the two solutions are the normal modes with corresponding angular frequencies, ω1 and ω2.
The (η1,η2) axes of the two normal modes correspond to a rotation of 45∘ in configuration space, Figure 14.3.1. The initial conditions chosen correspond to η1=−η2 and thus both modes are excited with equal intensity. Note that there are 5 lobes along the η2 axis versus 4 lobes along the η1 axis reflecting the ratio of the eigenfrequencies ω1 and ω2. Also note that the diamond shape of the motion in the (x1,x2) configuration space illustrates that the extrema amplitudes for x2 are a maximum when x1 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 (η1,η2) configuration space, the motion is bounded by a rectangle parallel to the (η1,η2) axes reflecting the fact that the extrema amplitudes, and corresponding energies, for the η1 normal mode are constant and independent of the motion for the η2 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 x1(0)=−x2(0), and ˙x1(0)=−˙x2(0), then η2(t)=0. That is, only the η1(t) normal mode is excited with frequency ω1 which corresponds to motion confined to the η1 axis of Figure 14.3.1.
As shown in Figure 14.3.2, η1(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 x1(0)=x2(0), and ˙x1(0)=˙x2(0), then η1(t)=0, that is, only the η2(t) normal mode is excited. The η2(t) normal mode is the symmetric mode where the two masses oscillate in phase with frequency ω2; it corresponds to motion along the η2 axis. For the symmetric phase, both masses move together leading to a constant extension of the coupling spring. As a result the frequency ω2 of the symmetric mode η2(t) is lower than the frequency ω1 of the asymmetric mode η1(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