Question

Let \( \omega_f \) be the excitation frequency of a sinusoidal load and \( \omega_n \) be the natural frequency of a single degree of freedom system. Then the dynamic response of the system is highly affected by the stiffness of the system when:

• A. \( \omega_f = \omega_n \)
• B. \( 0<\omega_f<\omega_n \)
• C. \( 0<\omega_n<\omega_f \)
• D. \( \omega_f = 0 \)

Hint:

Resonance occurs when the excitation frequency matches the natural frequency. The system's stiffness determines the amplitude of the response in this region.

Answer 1

The Correct Option is B

In a single degree of freedom (SDOF) system, the dynamic response is influenced by the relationship between the excitation frequency \( \omega_f \) and the natural frequency \( \omega_n \). The system's stiffness plays a critical role in determining how it responds to external loads. The dynamic response is most affected when: Step 1: Resonance occurs when the excitation frequency is equal to the natural frequency.
When \( \omega_f = \omega_n \), resonance occurs, and the system experiences large oscillations due to the large amplitude of the response.
Step 2: Stiffness effects when \( \omega_f \) is close to \( \omega_n \).
When \( \omega_f \) is close to \( \omega_n \) (i.e., \( 0<\omega_f<\omega_n \) or \( 0<\omega_n<\omega_f \)), the stiffness of the system significantly affects the response. The system may become more responsive to small perturbations in the excitation frequency, leading to large dynamic amplitudes. Thus, the correct answers are (B) and (C), as the stiffness of the system significantly affects the response when the excitation frequency is close to the natural frequency.