Re: Why the amplitude is only approximately greatest for resonance?

You ask “…what is the logic behind amplitude is in maximum when the driving frequency is the same as the natural one?”

In order for the driving energy to be most efficiently transferred into the system and create the greatest amplitude, the driving frequency must be the same as the natural resonant frequency, when there is no damping present. The reason is that the driving force must be “in phase”, or synchronized with the natural motion (frequency) of the system. An example is that if a person is on a swing and freely swinging back and forth, and you wish to push them higher, you must push at the correct time, or else you will be pushing against them and reducing their amplitude (height). If there is damping present in the system (and all real world systems have damping) this will reduce the “damped” resonate frequency so it will not be exactly the same as the natural resonant frequency. The larger the damping, the larger the effect and the lower the damped resonant frequency. Damping (such as shock absorber in an automobile suspension) reduces the amplitude of the oscillations every cycle until they reach zero amplitude (as can be seen with a bouncing ball). For further information see the link below:

See HyperPhysics - Simple Harmonic motion: http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html

See MathWorld - Simple Harmonic motion: http://mathworld.wolfram.com/SimpleHarmonicMotion.html

See MathWorld - Damped Simple Harmonic motion: http://mathworld.wolfram.com/DampedSimpleHarmonicMotion.html