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Driven Harmonic Motion

# Driven Harmonic Motion - an response would change with time...

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Driven Harmonic Motion The case of a harmonic oscillator driven by a sinusoidal varying force is an extremely important one in many branches of physics. In the previous sections we have discussed several examples of harmonic oscillators, and for each system we have been able to calculate the natural frequency [omega] 0 , (for example, for the spring [omega] 0 2 = k/m). The equation of motion for an oscillator on which no damping force is working, and no external force is applied is given by Suppose an external force F(t) is applied to this system. The external force has an amplitude m F 0 and an angular frequency [omega]. The equation of motion describing the system is now given by The steady state (the state of the system after any transient effects have died down) response of the system will be precisely at the driving frequency. Otherwise the relative phase between force
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Unformatted text preview: an response would change with time. Thus, the steady-state response of a harmonic oscillator is at the driving frequency [omega] and not at the natural frequency [omega] . The general solution of the equation of motion is Substituting this expression into the equation of motion we obtain This equation can be rewritten by using some trigonometric relations This equation can only be satisfied if the coefficients of cos([omega]t) and sin([omega]t) are zero. This implies that and In general A != 0 and [omega] != [omega] . The first condition than shows that The second condition can now be rewritten as The amplitude of the harmonic oscillator is given by The amplitude of the oscillation of the system gets very large if [omega] approaches [omega] . The system is said to be in resonance when this happens....
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