Steady state amplitude is given by

$\frac{X}{X_{st}} = \frac{1}{\sqrt{ 1 = (\frac{w}{w_n}^2] + [2 \xi \frac{w}{w_n}]^2}}$

Let $r = \frac{w}{w_n}$

$\frac{X}{X_{st}} = \frac{1}{\sqrt{ 1 – r^2)2 + (2 \ \xi r)^2}}$

Then for x to be maximum, the term $(1 – r^2)^2 + (2 \xi r)^2$ should be minimum.

$\therefore$ differentiate the above term with respect to r and equating it to zero.

Thus $\frac{d}{dr} [(1 – r^2)^2 + (2 \xi r)^2] = 0$

$\therefore$ $2(1-r^2)(-2r) + 4\xi r. 2r = 0$

$\therefore$ $1 – r^2 = 2 \xi^2$

$\therefore$ $r = \sqrt{1 – 2 \xi^2}$

$\frac{w_p}{w_n} = \frac{w_p}{w_n} = \sqrt{1 – 2 \xi^2}$

$\therefore$ $w_p = w_n \sqrt{1 – 2 \xi^2}$

Here $w_p$ is frequency corresponding to peak amplitude. Hence, from above equation, we can say that frequency at which peak amplitude occurs is just LHS of resonance on graph. From above equation, we can say no maxima or peak will occur when expression within radical sign is negative. Above equation will be negative when:

$\xi \ \gt \ \frac{1}{\sqrt{2}}$

i.e. $\xi \ \gt \ 0.707$

From the response curve, it is seen that, the response curve is always below the unity magnification line for $\xi \ \gt \ 0.707$

Phase lag v/s frequency ratio for different amounts of damps.

No peak will occur the response curve is always between unity magnification and we know that

$w_d \ = \ \sqrt{ 1 - \xi^2} \ w_n$

$w_d$ = damped natural frequency.

The phase angle also varies from zero at low frequencies to 180˚ at very high frequencies. It is 90 ˚ at resonance and is independent of damping. Over a small frequency range containing the resonance point. The variation of phase angle is more abrupt for lower values of damping than for higher values. More abrupt the change in phase angle about resonance. More sharp is the peak in the frequency response curve. For zero damping, the phase lag suddenly changes from zero to 180 ˚ at resonance. The corresponding zero damping frequency response cure is also infinitely sharp at resonance.