Hour 33 Coupled Oscillators I Physics 321 Hour 33 Coupled Oscillators I

Example I x1 x2 k1 k2 k3 𝑇= 1 2 𝑚 1 𝑥 1 2 + 1 2 𝑚 2 𝑥 2 2 𝑈= 1 2 𝑘 1 𝑥 1 2 + 1 2 𝑘 2 𝑥 2 − 𝑥 1 2 + 1 2 𝑘 3 𝑥 2 2 𝐿= 1 2 𝑚 1 𝑥 1 2 + 1 2 𝑚 2 𝑥 2 2 − 1 2 𝑘 1 𝑥 1 2 − 1 2 𝑘 2 𝑥 2 − 𝑥 1 2 − 1 2 𝑘 3 𝑥 2 2

Example I x1 x2 k1 k2 k3 𝐿= 1 2 𝑚 1 𝑥 1 2 + 1 2 𝑚 2 𝑥 2 2 − 1 2 𝑘 1 𝑥 1 2 − 1 2 𝑘 2 𝑥 2 − 𝑥 1 2 − 1 2 𝑘 3 𝑥 2 2 𝑚 1 𝑥 1 =− 𝑘 1 𝑥 1 + 𝑘 2 𝑥 2 − 𝑥 1 =− 𝑘 1 + 𝑘 2 𝑥 1 + 𝑘 2 𝑥 2 𝑚 2 𝑥 2 =− 𝑘 3 𝑥 2 − 𝑘 2 𝑥 2 − 𝑥 1 =− 𝑘 2 + 𝑘 3 𝑥 2 + 𝑘 2 𝑥 1

Example I x1 x2 k1 k2 k3 𝑚 1 0 0 𝑚 2 𝑥 1 𝑥 2 =− 𝑘 1 + 𝑘 2 − 𝑘 2 −𝑘 2 𝑘 2 + 𝑘 3 𝑥 1 𝑥 2 𝐌 𝑥 =−𝐊 𝑥

Normal Modes Assume solutions of the form: 𝑥 1 𝑡 = 𝐴 1 sin 𝜔𝑡+ 𝜑 1 Note that 𝜔 is the same in both equations! – This defines a “normal mode.” Then 𝐌 𝑥 =− 𝜔 2 𝐌 𝑥 =−𝐊 𝑥

Normal Modes - Method 1 𝐌 𝑥 =− 𝜔 2 𝐌 𝑥 =−𝐊 𝑥 → det 𝐊− 𝜔 2 𝐌 =0 𝐌 𝑥 =− 𝜔 2 𝐌 𝑥 =−𝐊 𝑥 → det 𝐊− 𝜔 2 𝐌 =0 𝑘 1 + 𝑘 2 − 𝜔 2 𝑚 1 − 𝑘 2 −𝑘 2 𝑘 2 + 𝑘 3 − 𝜔 2 𝑚 2 =0 Modified eigenvalue problem Solve for 𝜔 2 Use 𝐊− 𝜔 2 𝐌 𝑥 1 𝑥 2 =0 to solve for 𝑥 1 and 𝑥 2 Easiest by hand.

Normal Modes - Method 2 𝐌 𝑥 =− 𝜔 2 𝐌 𝑥 =−𝐊 𝑥 𝐌 −1 𝐊 𝑥 = 𝜔 2 𝑥 𝐌 𝑥 =− 𝜔 2 𝐌 𝑥 =−𝐊 𝑥 𝐌 −1 𝐊 𝑥 = 𝜔 2 𝑥 → det 𝐌 −1 𝐊− 𝜔 2 𝟏 =0 Standard eigenvalue problem Eigenvalues are 𝜔 2 Eigenvectors are normal modes Easiest with Mathematica

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