coup-osc - 8 Coupled Oscillators and Normal Modes Fall 2003...

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8 Coupled Oscillators and Normal Modes Fall 2003 An undamped harmonic oscillator (a mass m and a Hooke's-law spring with force constant k ) has only one natural frequency of oscillation, ϖ o = k m . But when two or more such oscillators interact, several natural frequencies are possible. Let's consider a system of masses and Hooke's-law springs that has a stable equilibrium position, such that each mass can vibrate around its equilibrium position. Are there possible motions in which every mass moves with simple harmonic motion, all masses with the same frequency? Such a motion, when it exists, is called a normal-mode motion. We will now develop general methods for finding the possible normal modes of such a system and their associated frequencies. We'll assume throughout that the spring forces are linear functions of displacement. We'll illustrate the general method by use of the following specific example. Example Let's consider the system shown below. The two masses move along a straight line. In the equilibrium positions, the springs are neither stretched nor compressed, and the coordinates x 1 and x 2 are the displacements of the particles from equilibrium. If k ' = 0 (i.e., if the center spring is removed), we have two uncoupled harmonic oscillators; each one can vibrate with angular frequency ϖ = k m , with arbitrary amplitude and phase. When the central spring is included, there are two cases where the masses can oscillate with the same frequency: 1) If x 1 = x 2 at each instant, then spring k ' is never stretched or compressed, and it can be ignored. The two masses vibrate sinusoidally, in phase, with the same angular frequency ϖ = k m , and with equal amplitudes. 2) If x 1 = - x 2 at each instant, the midpoint of spring k ' is stationary, and the force it exerts on each mass is like that of a spring with force constant 2 k '. The total force on each mass is the same as for a spring with force constant k + 2 k '. In this case, the two masses move sinusoidally with angular frequency ϖ = + ( ' ) k k m 2 , with equal amplitudes but a half-cycle out of phase. Thus this system has two
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This note was uploaded on 12/08/2011 for the course PHYSICS 340 taught by Professor Clarke during the Fall '08 term at University of Michigan.

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coup-osc - 8 Coupled Oscillators and Normal Modes Fall 2003...

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