Chapter 12 Coupled Oscillations

1. Chapter 12Coupled Oscillations Claude A Pruneau Wayne State University

2. 12.1 Introduction • Coupled equations • Normal coordinates • Normal modes • n degrees of freedom (n-coupled 1-d oscillators or n/3-coupled 3-d oscillators) leads to n normal modes (in general) • some of the modes may be identical.

3. 12.2 Two coupled harmonic oscillators. • Example: In a solid, atoms interact by elastic forces and oscillate about their equilibrium positions. • Let’s consider the following simpler system m1=M m2=M 1= 12 2= x2 x1

4. Consider a solution of the form Frequency, , to be determined, and amplitudes may be complex.

5. A solution to these Eqs exist if the 2x2 determinant is null. This yields

6. There are two characteristic frequencies The general solution is thus

7. The amplitudes are not all independent given that they must satisfy. The solutions may thus be written:

8. There are four arbitrary constants - as expected given one has two equations of second order.

9. add subtract

14. 1 f 0 2 1 2 = 0 0 3

15. Weak Coupling

18. X1(t) t X2(t) t

22. In general, we then find where By construction:

23. In summary

24. finite mij and Ajk express the coupling between the various coordinates.

26. Euler-Lagrange Eq.

28. We get A non trivial solution to this equation exists only if A secular Eq. of degree n in 2. Implies n roots for 2