Natural frequency of the sloshing water in a swimming pool of a cruiser ship is studied by an experimental set up with a scale of 1/10. If the natural frequency is found 3.5 Hz, then what is the natural frequency for full scale?
http://www.jiskha.com/display.cgi?id=1392897814
Sorry, I was rowing this morning and did not see this question. Finally a question I am used to :)
Naval Architect, Gloucester, MA
by the way if the wavelength is long compared to the water depth then the speed is proportional to the depth.
v = k L
T = L/v = L/kL = 1/k
and the speed would be so much faster in the deeper pool that the period would be the same.
sorry, have shallow water limit wrong
speed proportional to square root of depth
v = k L^.5
T = L/v = L/(k L^.5) = c sqrt L
so
frequency = constant / sqrt L
so same ratio, 1/sqrt 10
To determine the natural frequency for the full-scale swimming pool, we can use the principle of similarity. The frequency of an oscillating system is dependent on certain parameters such as mass, stiffness, and geometry. In this case, we want to determine the relationship between the natural frequency of the scale model and the full-scale prototype.
The scale factor, denoted as "S," relates the dimensions of the scale model to the full-scale prototype. In this case, the scale factor is given as 1/10, which means that all linear dimensions of the scale model are one-tenth of the corresponding dimensions of the full-scale prototype.
To establish the relationship between the natural frequencies of the scale model and the full-scale prototype, we need to consider the effect of the scale factor on the stiffness and mass of the system. The natural frequency (f) is inversely proportional to the square root of the effective mass (m_eff) and directly proportional to the square root of the effective stiffness (k_eff).
The formula for the natural frequency (f) is given by:
f = (1 / 2π) * sqrt(k_eff / m_eff)
Since both mass and stiffness are affected by the scale factor, we can express the relationship between the natural frequencies as follows:
f_full_scale = (1 / 2π) * sqrt(k_full_scale / m_full_scale)
f_scale_model = (1 / 2π) * sqrt(k_scale_model / m_scale_model)
Using the scale factor (S = 1/10), we can express the relationship between stiffness and mass for the scale model and full-scale prototype as:
k_scale_model = S^2 * k_full_scale
m_scale_model = S^3 * m_full_scale
Substituting these relationships into the formula for the natural frequency of the scale model:
f_scale_model = (1 / 2π) * sqrt((S^2 * k_full_scale) / (S^3 * m_full_scale))
Simplifying the equation:
f_scale_model = (1 / 2π) * sqrt(k_full_scale / (S * m_full_scale))
Given that the natural frequency for the scale model is 3.5 Hz, we can substitute this value into the equation and solve for the natural frequency of the full-scale prototype:
3.5 = (1 / 2π) * sqrt(k_full_scale / (S * m_full_scale))
Simplifying further:
3.5 * 2π = sqrt(k_full_scale / (S * m_full_scale))
Square both sides of the equation:
(3.5 * 2π)^2 = k_full_scale / (S * m_full_scale)
Now, plug in the values:
(3.5 * 2π)^2 = k_full_scale / (1/10 * m_full_scale)
Solve for k_full_scale:
k_full_scale = (3.5 * 2π)^2 * (1/10 * m_full_scale)
k_full_scale ≈ 77.22 * m_full_scale
Therefore, the natural frequency for the full-scale swimming pool on the cruiser ship would be approximately 77.22 times the natural frequency of the scale model, which is 3.5 Hz.