Question

A machine of mass 100 kg is subjected to an external harmonic force with a frequency of 40 rad/s. The designer decides to mount the machine on an isolator to reduce the force transmitted to the foundation. The isolator can be considered as a combination of stiffness (\(K\)) and damper (damping factor, \(\xi\)) in parallel. The designer has the following four isolators: 1) \(K = 640 \, \text{kN/m}, \, \xi = 0.70\)
2) \(K = 640 \, \text{kN/m}, \, \xi = 0.07\)
3) \(K = 22.5 \, \text{kN/m}, \, \xi = 0.70\)
4) \(K = 22.5 \, \text{kN/m}, \, \xi = 0.07\) Arrange the isolators in the ascending order of the force transmitted to the foundation.

• A. 1-3-4-2
• B. 1-3-2-4
• C. 4-3-1-2
• D. 3-1-2-4
  \textbf{Correct Answer:} (C)

Hint:

In vibration isolation problems, a balance between stiffness and damping is crucial. High damping reduces oscillations, but very high stiffness or damping alone may not always provide the best isolation. An optimal combination minimizes transmitted force effectively.

Answer 1

The Correct Option is C

The force transmitted to the foundation can be evaluated using the concept of vibration isolation. The isolation effectiveness depends on the resonant frequency and damping factor of the isolator. The transmitted force \(F_t\) is related to the following formula: \[ F_t = \frac{m \omega^2}{\sqrt{(K - m \omega^2)^2 + (c \omega)^2}} \] Where:
- \(m\) is the mass of the machine (100 kg),
- \(\omega\) is the angular frequency of the external harmonic force (40 rad/s),
- \(K\) is the stiffness of the isolator,
- \(c\) is the damping coefficient, which is related to the damping factor \(\xi\) by \(c = 2 \xi \sqrt{Km}\).
Step 1: Understand the Formula and Parameters To minimize the transmitted force, the isolator should have a balance between stiffness \(K\) and damping \(\xi\).
- Stiffness (\(K\)): Higher stiffness will lead to higher resonant frequencies, reducing the transmitted force.
- Damping (\(\xi\)): Higher damping reduces the amplitude of oscillations, which in turn reduces the force transmitted. However, too much damping can reduce the effectiveness of isolation at certain frequencies.
Thus, the ideal isolator will have a combination of high stiffness and damping to reduce the transmitted force effectively. Step 2: Analyze the Isolators Now, we need to arrange the isolators based on the values of \(K\) and \(\xi\). Let's analyze each option: - Option 1 (\(K = 640 \, \text{kN/m}, \, \xi = 0.70\)): This is a high stiffness isolator with a very high damping factor, leading to excellent isolation and low transmitted force.
- Option 2 (\(K = 640 \, \text{kN/m}, \, \xi = 0.07\)): While the stiffness is the same as option 1, the damping factor is much lower, which makes it less effective at reducing the transmitted force.
- Option 3 (\(K = 22.5 \, \text{kN/m}, \, \xi = 0.70\)): This isolator has a low stiffness value but a high damping factor. It will be effective in damping out oscillations, though the low stiffness may limit its ability to isolate effectively.
- Option 4 (\(K = 22.5 \, \text{kN/m}, \, \xi = 0.07\)): This has both low stiffness and low damping, making it the least effective in reducing transmitted force.
Step 3: Ascending Order of Force Transmitted From the above analysis, we can conclude that the isolators will transmit the least force in the following order: - Option 1 (high stiffness and high damping) will transmit the least force.
- Option 3 (high damping but low stiffness) will transmit a moderate amount of force.
- Option 2 (high stiffness but low damping) will transmit more force than option 3.
- Option 4 (low stiffness and low damping) will transmit the most force.
Thus, the correct order of isolators, from least to most transmitted force, is (C) 4-3-1-2.