Question

Consider a forced single degree-of-freedom system governed by \[ \ddot{x}(t) + 2 \zeta \omega_n \dot{x}(t) + \omega_n^2 x(t) = \omega_n^2 \cos(\omega t), \] where \( \zeta \) and \( \omega_n \) are the damping ratio and undamped natural frequency of the system, respectively, while \( \omega \) is the forcing frequency. The amplitude of the forced steady state response of this system is given by \[ \left[ (1 - r^2)^2 + (2 \zeta r)^2 \right]^{-1/2}, \quad \text{where} \quad r = \frac{\omega}{\omega_n}. \] The peak amplitude of this response occurs at a frequency \( \omega = \omega_p \). If \( \omega_d \) denotes the damped natural frequency of this system, which one of the following options is true?

• A. \( \omega_p<\omega_d<\omega_n \)
• B. \( \omega_p = \omega_d<\omega_n \)
• C. \( \omega_d<\omega_n = \omega_p \)
• D. \( \omega_d<\omega_n<\omega_p \)

Hint:

In a damped system, the resonant frequency \( \omega_p \) is less than the damped natural frequency \( \omega_d \), and \( \omega_d \) is less than the undamped natural frequency \( \omega_n \).

Answer 1

The Correct Option is A

For a forced single degree-of-freedom system, the frequency \( \omega_p \) at which the peak amplitude occurs is called the resonant frequency. This frequency is associated with the undamped natural frequency \( \omega_n \), and for a system with damping, the peak occurs slightly below \( \omega_n \). The damped natural frequency \( \omega_d \) is related to the undamped natural frequency \( \omega_n \) and the damping ratio \( \zeta \) by the following equation: \[ \omega_d = \omega_n \sqrt{1 - \zeta^2} \] Since \( \zeta \) is always less than 1 for a damped system, it follows that \( \omega_d \) is less than \( \omega_n \), but greater than \( \omega_p \), because the resonant frequency \( \omega_p \) occurs slightly below the undamped natural frequency. Thus, the correct relationship is: \[ \omega_p<\omega_d<\omega_n \] Therefore, the correct answer is (A). Final Answer:
\[ \boxed{(A)} \]