al phy

In a forced oscillation, energy is transfered from the driving force to the oscillating system. Damping leads to a dissipation of energy. Hence, amplitude of oscillation is limited by the damping force.

In resonace, the transfer of energy from the driving force to the oscillating system is a maximum. With the absence of anydamping force, very large amount of energy could be transferred to the system without any dissipation. Hence, the amplitude increases indefinitely.

要用式derive 出來的，是out of syllabus 的，都挺複雜。

Forced oscillation(Login required)We can describe such forced oscillation by harmonic function : Figure 1: The support of spring - block system in oscillation Forced oscillation x=Asin(ωet+φ) where “ ωe ” is the angular frequency of external force being applied on the system. Forced oscillationIt is clear from the discussion so far that most of artificial oscillation system tends to cease as damping is part of the natural set up. There can be various sources of damping force, but friction is one common source. There can be air resistance or resistance at the fixed hinge from which oscillating part is hung. It is imperative that we supply appropriate energy (force) to compensate for the loss of energy due to damping. To meet this requirement, the oscillating system is subjected to oscillatory external force. The external force imparted is itself oscillatory and is, therefore, described by harmonic trigonometric function. Considering presence of damping, the force equation is : *****************Fnet=−kx−bv+F0sinωet ma+kx+bv−F0sinωet=0 In terms of displacement derivatives : ⇒mⅆ2xⅆt2+bⅆxⅆt+kx−F0sinωet=0 The solution of this differential equation yields : x=Asin(ωet+φ) As is evident from the expression, the system oscillates with the same frequency as that of external force. The amplitude of the oscillation is described in terms of frequency of external force ( ωe) and natural frequency (ωo ) as : A=Fm{(ω2e−ω2o)+(bωem)2}−−−−−−−−−−−−−−−−−−−√ We can see that this expression is a constant for given set up. It means we can sustain a constant amplitude of the oscillation by applying external oscillatory force - even if damping force is present.