This is the original Excel file used to load data from HWs/HW2/turbine_tower_prob_ORIGINAL.xlsx
Using different shape functions an estimate of the natural frequency for the wind tower was found using the method of generalized single degree of freedom for each method.
The following table summarizes the results obtained. For each shape function the following items are calculated: Effective mass \(M_{e}\), effective stiffness \(K_{e}=K_{fe}+K_{ge},\)effective flexural stiffness \(K_{fe}\), effective geometric stiffness \(K_{ge},\) The ratio \(\frac {M}{M_{e}}\)and the natural frequency \(f\) in Hz.
The rows of the table below are listed from the lowest to the largest natural frequency found.
The shape function that produces the lowest natural frequency will be the one to select as the closest approximation to the real solution. The actual mass is \(404171\) Kg.
An Excel worksheet is also available on my web page for this HW for the lowest natural frequency case.
| shape function \(\Phi \left ( x\right ) \) | \(M_{e}\) kg) | \(K_{e}\) | Flexural \(K_{e}\ \left ( N/m\right ) \) | Geometric \(K_{ge}\) \(\left ( N/m\right ) \) | \(\frac {M_{e}}{M}\) | \(f\) (Hz) |
| \(\frac {x^{2}}{L^{2}}\) | \(159,636\) | \(383,031\) | \(393,520\) | \(-10489\) | \(39.49\%\) | \(0.2465\) |
| \(1-\cos \left ( \frac {\pi x}{2L}\right ) \) | \(164,157\) | \(431,388\) | \(441,587\) | \(-10198\) | \(40.62\) | \(0.2580\) |
| \(\frac {2Lx^{2}-x^{3}}{2L^{3}}\) | \(165,830\) | \(472,453\) | \(482,548\) | \(-10095\) | \(41.03\) | \(0.2686\) |
| first mode | \(168,445\) | \(543,282\) | \(553,333\) | \(-10051\) | \(41.68\) | \(0.2858\) |
| \(\frac {6L^{2}x^{2}-4Lx^{3}+x^{4}}{3L^{4}}\) | \(169,764\) | \(595,562\) | \(605,586\) | \(-10024\) | \(42\) | \(0.2981\) |
| 2nd mode | \(185,852\) | \(14,443,032\) | \(14,509,551\) | \(-66519\) | \(45.98\) | \(1.403\) |
| 3rd mode | \(192,575\) | \(100,304,976\) | \(100,475,002\) | \(-170026\) | \(47.65\) | \(3.6323\) |
| 4th mode | \(195,562\) | \(371,956,138\) | \(372,284,973\) | \(-328835\) | \(48.386\) | \(6.941\) |
The shape functions above indicated by the mode, are the mode shape function for a beam with fixed-free boundary conditions obtained from table \(8.1\) from reference [1].
The following diagram describes the computation done at each element of the wind tower
The lowest approximate natural frequency found is \(0.2465\) Hz for the shape function \(\frac {x^{2}}{L^{2}}\). The effective mass to actual mass ratio for this case was \(39.49\%\)
The higher the natural frequency became as the shape function is changed, this ratio also increased. At \(f=6.941\) Hz, this ratio became almost \(50\%\).
An applet was written to simulate the result allowing one to select different shape functions and observe the result.
This table shows the final computation result for the case that gave the lowest natural frequency
