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Unformatted text preview: Nanyang Technological University School of Physical and Mathematical Sciences Division of Physics and Applied Physics PAP 119 Physics Lab Ib Experiment 2: Coupled Oscillators Background A system of coupled oscillators is a system where the behavior of each variable inﬂuences that of the others, leading to a coupling of the oscillations of the individual degrees of freedom. In a system of coupled oscillators with N degress of freedom, there are N unique patterns of oscillation in which all masses oscillate at the same frequency with fixed amplitudes. These are called the normal modes . These modes are important, because any general motion of the coupled oscilaltors can be written as a linear combination of the normal modes, x ( t ) = N X i = 1 c i u i cos ω i t , (1) where u i denotes the i th normal mode. A single mass m connected by two springs with elastic constants k 1 = k 2 = k with one of the springs connected to a wave driver will experience a driving force of F ( t ) = F sin(2 π f t ) , (2) where F is the amplitude of the driving force. From Newton’s Second Law, we obtain the second- order di ff erential equation − 2 kx + F sin(2 π f t ) = m d 2 x dt 2 , (3) which has steady-state solution x ( t ) = A sin(2 π f t ) . (4) Here f is the driving frequency, and the steady-state amplitude...
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- Spring '10
- Normal mode, Nanyang Technological University, normal mode frequencies, Physics and Applied Physics