Question

Several bolts on the propeller of a fanboat detach, resulting in an offset moment of \(5 \mathrm{lb}-\mathrm{ft}\). Determine the amplitude of bobbing of the boat when the fan rotates at \(200 \mathrm{rpm}\), if the total weight of the boat and passengers is \(1000 \mathrm{lb}\) and the wet area projection is approximately \(30 \mathrm{sq} \mathrm{ft}\). What is the amplitude at $1000 \mathrm{rpm}$ ?

Short answer

Answer: The amplitude of bobbing at 200 rpm is 3.14 ft, and at 1,000 rpm is 15.71 ft.

Step-by-step solution

Calculate the oscillation frequency at each speed

First, we need to find the frequency of the oscillations caused by the fan's rotation. Frequency (\(f\)) is related to rotation speed (\(N\), in rpm) by the following formula: \(f = \frac{N}{60}\). Calculate the oscillation frequencies for \(200 \mathrm{rpm}\) and \(1,000 \mathrm{rpm}\): - For \(200 \mathrm{rpm}\): \(f_{200} = \frac{200}{60} = 3.33 \mathrm{Hz}\) - For \(1,000 \mathrm{rpm}\): \(f_{1000} = \frac{1000}{60} = 16.67 \mathrm{Hz}\)

Find the equivalent spring constant of the boat's buoyancy

The equivalent spring constant of the boat's buoyancy (\(k\)) can be found using the wet area projection (\(A\), in sq ft) and the weight of the boat and passengers (\(W\), in lb): \(k = \frac{W}{A}\) \(k = \frac{1000}{30} = 33.33 \mathrm{lb/ft}\)

Calculate the damping coefficient

The damping coefficient (\(c\)) can be found using the moment (\(M\), in lb-ft) and the oscillation frequency (\(f\), in Hz): \(c = 2\pi f M\) Calculate the damping coefficients for the two oscillation frequencies we found in Step 1: - For \(3.33 \mathrm{Hz}\): \(c_{200} = 2\pi (3.33)(5) = 104.72 \mathrm{lb\cdot s/ft}\) - For \(16.67 \mathrm{Hz}\): \(c_{1000} = 2\pi (16.67)(5) = 523.60 \mathrm{lb\cdot s/ft}\)

Determine the amplitude of bobbing at each speed

The amplitude of bobbing (\(A_b\), in ft) can be found using the damping coefficient (\(c\), in lb·s/ft) and the equivalent spring constant (\(k\), in lb/ft) by the following formula: \(A_b = \frac{c}{k}\): - For \(200 \mathrm{rpm}\): \(A_{b200} = \frac{104.72}{33.33} = 3.14 \mathrm{ft}\) - For \(1,000 \mathrm{rpm}\): \(A_{b1000} = \frac{523.60}{33.33} = 15.71 \mathrm{ft}\) So, the amplitude of bobbing at \(200 \mathrm{rpm}\) is \(3.14 \mathrm{ft}\) and at \(1,000 \mathrm{rpm}\) is \(15.71 \mathrm{ft}\).