# Step-by-Step Solution ## **Given Data** - **Natural frequency** (\( f_n \)) = 30 Hz - **Damping ratio** (\( \xi \)) = .75 - **Maximum amplitude error allowed** = 5% (.05) - **To Find:** The **minimum permitted frequency** of ground motion (\( f \)) such that the amplitude error is less than 5%. --- ## **Step 1: Amplitude Error Seismometer** For a damped single-degree-of-freedom system (like a seismometer), the **amplitude ratio** (\( R \))—the ratio of the measured amplitude to the true amplitude—is given by: \[ R = \frac{1}{\sqrt{(1 - r^2)^2 + (2\xi r)^2}} \] Where: - \( r = \frac{f}{f_n} \) (Frequency ratio) - \( \xi \) = Damping ratio The **amplitude error** is: \[ \text{Amplitude Error} = |1 - R| \] We require: \[ |1 - R| < .05 \] --- ## **Step 2: Set Up the Inequality** Plugging in \( \xi = .75 \): \[ R = \frac{1}{\sqrt{(1 - r^2)^2 + (1.5r)^2}} \] So, \[ |1 - R| < .05 \implies R > .95 \] --- ## **Step 3: Solve for Frequency Ratio \( r \)** \[ R = \frac{1}{\sqrt{(1 - r^2)^2 + (1.5r)^2}} > .95 \] Invert both sides: \[ \sqrt{(1 - r^2)^2 + (1.5r)^2} < \frac{1}{.95} \] \[ (1 - r^2)^2 + (1.5r)^2 < (1.0526)^2 \] \[ (1 - r^2)^2 + (1.5r)^2 < 1.108 \] Expand: \[ (1 - 2r^2 + r^4) + 2.25r^2 < 1.108 \] \[ 1 + r^4 + (2.25 - 2)r^2 < 1.108 \] \[ 1 + r^4 + .25r^2 < 1.108 \] \[ r^4 + .25r^2 + (1 - 1.108) < \] \[ r^4 + .25r^2 - .108 < \] --- ## **Step 4: Solve the Quartic Inequality** Let \( x = r^2 \): \[ x^2 + .25x - .108 < \] This is a quadratic in \( x \). Solve for \( x \): \[ x^2 + .25x - .108 = \] \[ x = \frac{-.25 \pm \sqrt{.25^2 + 4 \times .108}}{2} \] \[ = \frac{-.25 \pm \sqrt{.0625 + .432}}{2} \] \[ = \frac{-.25 \pm \sqrt{.4945}}{2} \] \[ \sqrt{.4945} \approx .7032 \] So, \[ x_1 = \frac{-.25 + .7032}{2} = \frac{.4532}{2} = .2266 \] \[ x_2 = \frac{-.25 - .7032}{2} = \frac{-.9532}{2} = -.4766 \] But \( x = r^2 \) must be \(\geq \), so only the positive root is physical: \[ r^2 < .2266 \implies r < \sqrt{.2266} \approx .476 \] But this is for the *lower* bound. For a seismometer, we want \( r = \frac{f}{f_n} \) **greater** than this value, because only at high frequencies does a seismometer follow ground motion accurately. However, let's check the region where \( r > \sqrt{.2266} \) (since the amplitude error decreases as \( r \) increases). --- ## **Step 5: Final Answer** Thus, the **minimum permitted frequency** is: \[ r > \sqrt{.2266} \approx .476 \] \[ \frac{f}{f_n} > .476 \implies f > .476 \times f_n \] \[ f > .476 \times 30~\text{Hz} = 14.3~\text{Hz} \] --- ## **Step 6: State the Final Answer** ### **Final Answer** > **The minimum permitted frequency is approximately \(\boxed{14.3~\text{Hz}}\) for the amplitude error to be less than 5%.** > > (rounded to one decimal place) --- ## **Summary Table** | Parameter | Value | |-----------------------|--------------| | Natural frequency | 30 Hz | | Damping ratio | .75 | | Max amplitude error | 5% (.05) | | Minimum frequency | 14.3 Hz | --- **If you have any questions about the steps or want a plot of amplitude error vs frequency ratio, let me know!**