physics SHM resonance

Q: why the natural frequency is a straight line?
The natural frequency is a fixed value which depends on the characteristics of the oscillating system. The dotted-line on the graph only indicates to you the position on the frequency axis where the natural frequency is located.

Q: to make resonance occur, the driving frequency should close to
or equal to the natural frequency in each case?
For resonance occurs, the driving frequency from the source should be equal to the natural frequency of the oscillating system. But in case where damping occurs, because damping dissipates energy, the resonsance peak wil occur at a a frequency lower than the natural frequency.

The following web-page gives a good animation to a forced oscillation system. You could vary the damping (i.e. attenuation) and watch the result.

http://www.walter-fendt.de/ph14e/resonance.htm

Instead of dwelling on a qualitative analysis (couldn't give a better one), here is the exact results that answer your questions: (1) the amplitude-frequency dependence, (2) resonance frequency vs natural frequency. I also include an expression for phase shift.


圖片參考：http://imgcld.yimg.com/8/n/HA05726829/o/701007130145113873368570.jpg


Eq.(3) answer the first question. And here is a plot of amplitude against frequency based on eq.(3)


圖片參考：http://imgcld.yimg.com/8/n/HA05726829/o/701007130145113873368571.jpg


where the dotted line is at the natural frequency. Here larger resistance corresponds to smaller peak amplitude.

Eq.(4) gives the expression for resonance frequency. This shows it is always smaller than the natural frequency. And larger the resistance, the more the resonance frequency moves away from the natural frequency.

Finally here is a plot of the phase lag from eq.(3), the alpha expression.


圖片參考：http://imgcld.yimg.com/8/n/HA05726829/o/701007130145113873368582.jpg

Here the step is at the natural frequency, and is for a non-resistive system. And the phase shift become smoother as resistance gets stronger (larger b). Note that the phase shift collapses at pi/2 at the natural frequency (not the resonance frequency) for all resistance.