Understanding Machine Component Relative Displacement

This problem involves calculating the maximum relative displacement of a machine component when it undergoes harmonic base motion. We are given the component's natural frequency, the acceleration of the base motion, and the frequency of the base motion. Crucially, the damping is stated to be negligible.

Given Parameters

• Natural frequency, $ \omega_n = 20 $ rad/s
• Base motion acceleration amplitude, $ A_{base} = 3 $ m/s$^2$
• Base motion frequency, $ \omega = 10 $ rad/s
• Damping ratio, $ \zeta = 0 $ (negligible damping)

Concept of Base Excitation and Relative Displacement

When the base of a vibrating system moves, it causes the system's mass to experience inertial forces. The relative displacement is the motion of the mass with respect to its base. For a system subjected to base acceleration $ \ddot{y}(t) $, the equation of motion for the relative displacement $ x(t) $ (where $ x = u - y $, $ u $ is absolute displacement of mass, $ y $ is base displacement) with negligible damping is:

$$ \ddot{x} + \omega_n^2 x = -\ddot{y}(t) $$

Here, $ \ddot{y}(t) $ represents the base acceleration, which is given as harmonic. The amplitude of this base acceleration is $ A_{base} $. So, we can write $ \ddot{y}(t) = A_{base} \cos(\omega t) $.

Formula for Relative Displacement Amplitude

For a system with base acceleration $ \ddot{y}(t) = A_{base} \cos(\omega t) $ and negligible damping ($ \zeta = 0 $), the amplitude of the relative displacement $ X $ is given by the formula:

$$ X = \frac{A_{base}}{|\omega_n^2 - \omega^2|} $$

Calculating Peak Relative Displacement Amplitude

Step-by-Step Calculation

Now, let's substitute the given values into the formula:

1. Calculate the square of the natural frequency: $ \omega_n^2 = (20 \text{ rad/s})^2 = 400 \text{ rad}^2/\text{s}^2 $
2. Calculate the square of the base motion frequency: $ \omega^2 = (10 \text{ rad/s})^2 = 100 \text{ rad}^2/\text{s}^2 $
3. Calculate the difference between the squared frequencies: $ \omega_n^2 - \omega^2 = 400 \text{ rad}^2/\text{s}^2 - 100 \text{ rad}^2/\text{s}^2 = 300 \text{ rad}^2/\text{s}^2 $
4. Take the absolute value of the difference: $ |\omega_n^2 - \omega^2| = |300 \text{ rad}^2/\text{s}^2| = 300 \text{ rad}^2/\text{s}^2 $
5. Calculate the relative displacement amplitude $ X $ using the formula: $ X = \frac{A_{base}}{|\omega_n^2 - \omega^2|} = \frac{3 \text{ m/s}^2}{300 \text{ rad}^2/\text{s}^2} $
6. Perform the division: $ X = 0.01 \text{ m} $

Final Result Conversion

The question asks for the amplitude in millimeters (mm). To convert meters to millimeters, we multiply by 1000:

$$ X = 0.01 \text{ m} \times \frac{1000 \text{ mm}}{1 \text{ m}} = 10 \text{ mm} $$

Conclusion

Therefore, the peak amplitude of the relative displacement of the component, with negligible damping, is 10.0 mm.