thank you..
the frquecy response would then be shown with the bell curve right ? with max. amplitude at the natural freq. of the system.
but what would be the determinants of the shapness of resonance?
thank you once again
-apache
The determinants of the sharpness in resonance depends on the system involved. Lots of physical systems undergo resonance peaks, in the case of inhomogenously broadened optical oscillators (lasers) for example, the sharpness depends on the losses in the cavity, the gain medium, the shape and reflectivity of the mirrors and the length of the cavity.
Resonance peaks can be described mathematically using Gaussian, Lorentzian or a combination between the two (convolution) called a Voight profile. All these mathematicals functions have 2 important parameters, the maximum value and the Full width-Half maximum (The width of the curve at 1/2 the maximum value.) The ratio of these two parameters gives the Q values. Tall skinny peaks have high a Q, flat broad peaks have a low Q.
As enigma correctly points out, there need not be a resonant peak at all. Indeed there may be many overlapping resonance peaks (common in the case of lasers).
thanks for the replies,
yeah i was wrong on the bell curve lol .. but it kinda looks like a bell, with 0 gradient and max amplitude at f0 ...
anyhow.. i dont think i understand how the sharpness would be affected in mechanical SHM .. like mabye a spring and mass etc..
thanks for all your help
Resonance peaks do not occur in Simple Harmonic oscillators, only in mechanical oscillators with some damping constant, r and some sinusoidal driving force, F(t).
The frequency of oscillation you get with SHM is called the natural resonance of the system, so called because it resonates at that frequency without a driving force (i.e. F(t)=0). For damped oscillators, the natural frequency is slightly different than that of an undamped system (i.e r=0, as in SHM).
It is possible to force an oscillator to oscillate at a frequency other than its natural frequency using a driving force, however the amplitude decreases. Plotting the driving force frequency vs amplitude will give a resonance peak.
Some characteristics of mechanical resonance peaks include:-
- Amplitude = F/s as w (drive frequency) approaches 0 (Hooke's Law)
- Amplitude approaches 0 as w approaches infinity.
Resonance does occur in simple harmonic oscillator, resonance curve is then a delta-function (zero width and infinite amplitude).
Equation of forced oscillations in oscillating system can be written as: x"+(w_{o}/Q)x'+w_{o}^{2}=w_{o}^{2}cos(wt), where w_{o} is own (resonance) frequency of system and w - frequency of external force, 1/Q - damping factor (Q is usually called quality of oscillating system).
Stationary solution of this equation (established oscillations in such system after some time) is: x(t)=x_{o}cos(wt), where amplitude of oscillations depends on frequency w of external force: x_{o}={[(1-(w/w_{o})^{2}]^{2}+(w/Qw_{o})^{2}}^{-1}
You may see that for ideal oscillator (Q=oo) amplitude x_{o} becomes delta function. For non-infinite Q values the width of resonance curve deltaw/w_{o} (FWHM = width on 1/2 level for energy = width on 0.71 level for amplitude) is equal to 1/Q.
Typical Q values: mass on spring 1-10, radio LC cirquit 10-100, mass on a string 100-1000, quarts watch crystal 10^{4-5}, spectral lines (in visible range) 10^{5-6}, orbiting moons and planets 10^{7-9}, good laser 10^{8-9}, atomic clock cavity 10^{11-13}.
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