Exercise 5.4: superposition of motion

The results of the previous exercises make plausible the assumption that an arbitrary motion of the system can be written as

$$u_1(t) = A_1\cos{(\omega _1t+\delta _1)} + A_2\cos{(\omega _2 t+\delta _2)}$$ (50)

$$u_2(t) = A_1\cos{(\omega _1t+\delta _1)} - A_2\cos{(\omega _2 t+\delta _2)}$$ (51)

The values of these constants can be expressed in terms of the initial values of the displacement and velocities of each particle. determine these constants for $u_{1}=0.5,u_{2}=0,v_{1}=v_{2}=0$. Verify that the motion predicted by these equations is consistent with your measure values for $% u_{1}$ and $u_{2}$ for $k=1$ and $k_{c}=0.8$ in part 1 of Exercise 5.2. What is the periodicity of $u_{1}$ and $u_{2}$?

The effect of the spring $k_{c}$ is to couple the motions of two particles so that they no longer move independently. For special initial conditions, only one frequency of oscillation appears. The resulting motion is called a ``normal mode'' of the system, and the corresponding frequency a ``normal mode frequency''. The higher frequency is given by $\omega _{1}^{2}=(k+2k_{c})/m$. In this mode, the two particles oscillate exactly out of phase with the displacements of opposite directions. The motion at the lowest frequency $\omega _{2}^{2}=k/m$ corresponds to the two particles oscillating exactly in phase.

The general motion is a superposition of the two normal modes. Unless there is a simple relation between the two frequencies, the general motion is a complicated function of time. However, if the coupling is small, $\omega _{1}$ and $\omega _{2}$ are nearly equal and $u_{1}$ and $u_{2}$ exhibit ``beats''. In this case the displacements oscillate rapidly at the angular frequency $% 1/2(\omega _{1}+\omega _{2})$ with an amplitude that varies sinusoidally at the beat frequency $1/2(\omega _{1}-\omega _{2})$.

We also found that if we drive the system by an external force applied to either particle (or both), the system is in resonance if the frequency of the force corresponds to either of the normal modes. We use this method for determining the normal mode frequencies in the following exercises.