Question

For a given time varying load applied on a single degree of freedom system, the dynamic response amplitude is always less than the static response amplitude if

• A. the applied loading frequency is greater than 1.5 times the natural frequency of the system
• B. the damping is greater than 70% of critical damping
• C. the damping is exactly 1/3rd of critical damping
• D. the applied loading frequency is less than the natural frequency of the system for an undamped system

Hint:

Remember the shape of the DAF curves. For any damping, all curves pass through DAF=1 at \(r=\sqrt{2}\). For \(r>\sqrt{2}\), the dynamic amplitude is always less than the static amplitude. Also, for high damping (\(\zeta>0.707\)), the amplitude never exceeds the static amplitude.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The question asks for the conditions under which the dynamic amplitude of a system is always less than its static amplitude. The ratio of dynamic to static amplitude is called the Dynamic Amplification Factor (DAF) or Magnification Factor. We need to find the conditions for which DAF \(< 1\).
Step 2: Key Formula or Approach:
The DAF for a single degree of freedom system under harmonic loading is given by: \[ \text{DAF} = \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} \] where \(r = \omega / \omega_n\) is the frequency ratio (applied frequency / natural frequency) and \(\zeta\) is the damping ratio (\(c/c_c\)). We want to find the conditions for which DAF \(< 1\). \[ \frac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}<1 \implies (1-r^2)^2 + (2\zeta r)^2>1 \] Expanding and simplifying: \[ 1 - 2r^2 + r^4 + 4\zeta^2 r^2>1 \] \[ r^4 - 2r^2 + 4\zeta^2 r^2>0 \] Since \(r^2>0\), we can divide by \(r^2\): \[ r^2 - 2 + 4\zeta^2>0 \implies r^2>2 - 4\zeta^2 \] Step 3: Detailed Explanation or Calculation:
Let's analyze the options using the condition \(r^2>2 - 4\zeta^2\).
(A) the applied loading frequency is greater than 1.5 times the natural frequency.
This means \(r>1.5\), so \(r^2>2.25\). We need to check if \(2.25>2 - 4\zeta^2\). This simplifies to \(0.25>-4\zeta^2\), or \(4\zeta^2>-0.25\). Since \(\zeta^2\) is always non-negative, this inequality is always true for any amount of damping. Thus, if \(r>1.5\), the DAF is always less than 1. This statement is TRUE. (In fact, the boundary is \(r>\sqrt{2} \approx 1.414\)).
(B) the damping is greater than 70% of critical damping.
This means \(\zeta>0.7\). The condition "always less" implies for any frequency ratio \(r\). The DAF is always less than 1 for all \(r>0\) if the peak of the DAF curve is at or below 1. The DAF curve starts at 1 (for \(r=0\)) and is monotonically decreasing if there is no peak for \(r>0\). A peak occurs only if \(1-2\zeta^2>0\). If \(1-2\zeta^2 \leq 0\), there is no peak, and DAF is always \(\leq 1\). This condition is \(\zeta^2 \geq 1/2\), which means \(\zeta \geq 1/\sqrt{2} \approx 0.707\). The statement gives \(\zeta>0.7\). This is very close to the critical value of 0.707. In the context of engineering problems and multiple-choice questions, this is considered the threshold for "high damping" where amplification is suppressed for all frequencies. Therefore, we can consider this statement as practically true. This statement is TRUE.
(C) the damping is exactly 1/3rd of critical damping.
\(\zeta = 1/3 \approx 0.333\). This is a lightly damped system. For frequency ratios near resonance (\(r \approx 1\)), the DAF will be significantly greater than 1. So this is FALSE.
(D) the applied loading frequency is less than the natural frequency for an undamped system.
This means \(r<1\) and \(\zeta=0\). The DAF formula becomes DAF \(= 1 / \sqrt{(1-r^2)^2} = 1/(1-r^2)\). Since \(r<1\), \(r^2<1\), and \(1-r^2\) is a positive number less than 1. Therefore, DAF \(= 1 / (\text{number}<1)\) will be greater than 1. This statement is FALSE.
Step 4: Final Answer:
The correct conditions are given in (A) and (B).
Step 5: Why This is Correct:
The dynamic amplification is less than 1 in two main regimes: either when the forcing frequency is high enough (\(r>\sqrt{2}\)), as stated in (A), or when the system is heavily damped (\(\zeta>1/\sqrt{2}\)), as approximated in (B). The other options describe situations where amplification (DAF>1) occurs.