What is actually a resonating vibration and resonance? What is actually a resonating vibration and resonance?
I have searched many books and made Google search too but couldn't understand it clearly.
 A: Consider a pendulum, such as a playground swing.
It is a 2nd-order system, because it swings in a repetitive motion at a certain frequency like, say, 30 swings per minute.
It also is damped, because if you set it swinging and then leave it alone, it rubs against the air and its swings becomes smaller and smaller until it seems to have stopped.
Now, if you give a shove (put energy into it) at the same frequency that it swings (30 times per minute), the swings will get larger and larger until they get really large.
That's what happens when you drive a 2nd-order system at its resonant frequency.
A: Mechanical systems have properties such as mass and stiffness.
There is another property derived from the first two which is called natural frequency.
In general, these mechanical system can have various natural frequencies. But lets assume the representation of a single natural frequency that can be written as
$$\omega_n = \sqrt{\frac{k}{m}}$$, with units in [rad/seconds].
where $k$ is the stiffness coefficient and $m$ is the mass.
Resonance is when an external dynamic force of the type $f(t) = F\cos{(\omega t)}$ excites the system with the forcing frequency $\omega$ being very close to the natural frequency $\omega_n$.
A: A classical harmonic oscillator follows the differential equation:
$$A\frac{d^2x}{dt^2} = -Bx$$
The solution is a sinusoidal function as:
$x = K_1sin(\omega t + K_2)$
$K_1$ is the amplitude and $K_2$ the phase of the oscillation, while $$\omega = \sqrt{\frac{B}{A}}$$ is the frequency.
Note that the amplitude is not determined, and theoretically any value satisfies the equation. Of course there are physical limitations. For a system spring-mass, the limitation is the elastic range of the spring. But under that restrictions, the frequency $\omega$ of the oscillations is the same for any amplitude.
But that is an ideal case. Real oscillators have same damping (as air drag for example). So, for a system mass-spring, during any portion of the oscillation, besides the restoring force, there is also a drag force proportional to velocity and pointing against its direction.
The equation becomes:
$$A\frac{d^2x}{dt^2} = - Bx - C\frac{dx}{dt}$$
Now suppose that there is an external force, oscillating with the same frequency and in the same direction of the velocity. But with $90^\circ$ out of phase, because we need maximum force when the velocity is maximum, and minimum force when it is minimum. The equation becomes:
$$A\frac{d^2x}{dt^2} = - Bx - C\frac{dx}{dt} + D\cos((\omega t + K_2))$$
The solution is the same:
$x = K_1sin(\omega t + K_2)$
Because, on substituting on the equation, only the 2 last terms remains:
$$C\frac{dx}{dt} = D\cos((\omega t + K_2))$$
Substituting the solution:
$\omega K_1Ccos(\omega t + K_2) = D\cos((\omega t + K_2))$
It is fulfilled when:
$$K_1 = \frac{D}{\omega C}$$
Now the amplitude is determined, and is of course smaller for big damping coeficients, and bigger when the amplitude of the applied force is bigger.
The idea behind ressonance is to get a complete compensation of the damping by an external oscillating source.
