Forced Single Degree-of-Freedom System Analysis

The question asks us to determine the correct relationship between three important frequencies in a forced single degree-of-freedom system:

• Undamped Natural Frequency (\(\omega_n\)): This is the frequency at which the system would oscillate if there were no damping and no external forces. It is inherent to the system's mass and stiffness properties.
• Damped Natural Frequency (\(\omega_d\)): This is the frequency at which the system oscillates when disturbed and allowed to vibrate freely under damped conditions (transient response).
• Peak Amplitude Frequency (\(\omega_p\)): This is the forcing frequency at which the amplitude of the forced steady-state response reaches its maximum value. For damped systems, this is often referred to as the resonant frequency.

Understanding Key Frequencies in Damped Systems

Let's first define the relevant frequencies based on standard vibration theory:

1. Undamped Natural Frequency (\(\omega_n\)):

   From the given differential equation \(\rm \ddot x(t) + 2 \zeta \omega_n \dot x (t) + \omega_n^2 x(t) = \omega_n^2 \cos (\omega t)\), the undamped natural frequency is directly identified as \(\omega_n\).

2. Damped Natural Frequency (\(\omega_d\)):

   The damped natural frequency is given by the formula:

   \[ \omega_d = \omega_n \sqrt{1 - \zeta^2} \]

   Here, \(\zeta\) is the damping ratio. For a system to oscillate (underdamped), \(\zeta\) must be less than 1 (\(\zeta < 1\)).

3. Peak Amplitude Frequency (\(\omega_p\)):

   This is the frequency at which the amplitude of the steady-state response is maximum. The problem provides the amplitude of the forced steady-state response as:

   \[ A = [(1 - r^2)^2 + (2\zeta r)^2]^{-1/2} \]

   where \(r = \omega/\omega_n\). To find the frequency at which this amplitude is maximum, we need to find the value of \(r\) that maximizes \(A\). Maximizing \(A\) is equivalent to minimizing its denominator, \(f(r) = (1 - r^2)^2 + (2\zeta r)^2\).

Deriving Peak Amplitude Frequency (\(\omega_p\))

To find the value of \(r\) that minimizes \(f(r)\), we differentiate \(f(r)\) with respect to \(r\) and set the derivative to zero:

Let \(f(r) = (1 - r^2)^2 + (2\zeta r)^2\)

Expand the terms:

\[ f(r) = (1 - 2r^2 + r^4) + (4\zeta^2 r^2) \] \[ f(r) = r^4 + (4\zeta^2 - 2)r^2 + 1 \]

Now, differentiate \(f(r)\) with respect to \(r\):

\[ \frac{df(r)}{dr} = 4r^3 + 2(4\zeta^2 - 2)r \]

Set the derivative to zero to find the critical points:

\[ 4r^3 + (8\zeta^2 - 4)r = 0 \] \[ 4r(r^2 + 2\zeta^2 - 1) = 0 \]

Since \(r = \omega/\omega_n\), and for a non-zero resonant frequency, \(r \neq 0\), we consider the second factor:

\[ r^2 + 2\zeta^2 - 1 = 0 \] \[ r^2 = 1 - 2\zeta^2 \]

Since \(r = \omega_p/\omega_n\) at peak amplitude, let's denote this specific \(r\) as \(r_p\):

\[ r_p = \sqrt{1 - 2\zeta^2} \]

Substituting \(r_p = \omega_p/\omega_n\):

\[ \frac{\omega_p}{\omega_n} = \sqrt{1 - 2\zeta^2} \] \[ \omega_p = \omega_n \sqrt{1 - 2\zeta^2} \]

For \(\omega_p\) to be a real frequency, the term inside the square root must be non-negative: \(1 - 2\zeta^2 \ge 0\), which implies \(\zeta^2 \le 1/2\), or \(\zeta \le 1/\sqrt{2}\) (\(\approx 0.707\)). If \(\zeta\) is greater than \(1/\sqrt{2}\), the maximum amplitude occurs at \(\omega = 0\) (static deflection).

Comparing Damped Natural Frequency (\(\omega_d\)) and Undamped Natural Frequency (\(\omega_n\))

We have \(\omega_d = \omega_n \sqrt{1 - \zeta^2}\).

For any real damped system, the damping ratio \(\zeta\) is positive (\(\zeta > 0\)) and typically less than 1 for underdamped systems. Therefore, \(\zeta^2 > 0\).

This implies:

\[ 1 - \zeta^2 < 1 \] \[ \sqrt{1 - \zeta^2} < 1 \]

Multiplying by \(\omega_n\) (which is positive):

\[ \omega_n \sqrt{1 - \zeta^2} < \omega_n \] \[ \omega_d < \omega_n \]

Comparing Peak Amplitude Frequency (\(\omega_p\)) and Damped Natural Frequency (\(\omega_d\))

We have \(\omega_p = \omega_n \sqrt{1 - 2\zeta^2}\) and \(\omega_d = \omega_n \sqrt{1 - \zeta^2}\).

Since \(\zeta > 0\), it follows that \(2\zeta^2 > \zeta^2\).

This means:

\[ 1 - 2\zeta^2 < 1 - \zeta^2 \]

Taking the square root (and assuming \(\zeta \le 1/\sqrt{2}\) so that both terms are real):

\[ \sqrt{1 - 2\zeta^2} < \sqrt{1 - \zeta^2} \]

Multiplying by \(\omega_n\):

\[ \omega_n \sqrt{1 - 2\zeta^2} < \omega_n \sqrt{1 - \zeta^2} \] \[ \omega_p < \omega_d \]

Final Relationship Between Frequencies

Combining the inequalities we derived:

• \(\omega_d < \omega_n\)
• \(\omega_p < \omega_d\)

Therefore, the overall relationship is:

\[ \omega_p < \omega_d < \omega_n \]

This relationship holds true for underdamped systems with damping ratios \(\zeta \le 1/\sqrt{2}\), which is the usual context for such problems to have a distinct resonant peak.

Based on our derivation and comparison, the correct option is \(\omega_p < \omega_d < \omega_n\).