Question 4.6: A SDOF system has a mass of 100 kg, stiffness of 100 kN/m an...
A SDOF system has a mass of 100 kg, stiffness of 100 kN/m and a damping of 10 per cent. It is subjected to a force as shown in Figure 4.9. Period of the forcing function is 0.12 sec and its duration is also 0.12 sec. Determine the Fourier constants of force and displacement up to 20 terms. Also, plot the steady state response.
Natural frequency of SDOF = 31.623 rad/sec
Forcing frequency = 2Ï€/0.12 = 52.36 rad/sec
A computer code was written in FORTRAN using Equations (4.2) and (4.7), and the results are shown in Table 4.1.
 a_{n}=\frac{2}{T_{p}} \int_{0}^{T_{p}}p(t)\cos \frac{2\pi n}{T_{p}} t     n = 1, 2, 3….       (4.2)
 x(t)=\frac{1}{k} (a_{0}+\sum\limits_{n=1}^{\infty } \frac{1}{(1-\beta ^{2}_{n} )^{2}+(2\xi \gamma _{n})^{2}}
{\left[a_{n}2\xi \beta _{n}+b_{n}(1-\beta _{n}^{2})\right]\sin n \bar{\omega }_{1}t
+\left[a_{n}(1-\beta _{n}^{2})-b_{n}2\xi \beta _{n}\right]\cos n \bar{\omega }_{1} t }Â Â Â Â Â Â Â Â (4.7)
Fourier force coefficient A_{0} = 30000.000
Fourier displacement coefficient X_{0} = 0.300
The plot of the Fourier periodic force due to first 20 terms is shown in Figure 4.10 and that of the response in Figure 4.11.
| Fourier Constants for Force and Displacement Terms | ||||||
| Fourier Force Coeff. |
Fourier Displacement Coeff. |
|||||
| N | Frequency | Frequency Ratio | A(N) COS term |
B(N) SINE term | X1(N) COS term |
X2(N) SINE term |
| 1 | 52.36 | 1.656 | −2.13E − 03 | 4.86E + 04 | −2.70E − 01 | −5.12E − 02 |
| 2 | 104.72 | 3.312 | −2.43E + 04 | −2.13E − 03 | −1.61E − 03 | 2.43E − 02 |
| 3 | 157.08 | 4.967 | 7.09E − 04 | −5.40E + 03 | 2.28E − 03 | 9.56E − 05 |
| 4 | 209.44 | 6.623 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 |
| 5 | 261.799 | 8.279 | −4.25E − 04 | 1.95E + 03 | −2.88E − 04 | −7.06E − 06 |
| 6 | 314.159 | 9.935 | −2.70E + 03 | −7.09E − 04 | −5.62E − 06 | 2.76E − 04 |
| 7 | 366.519 | 11.59 | 3.04E − 04 | −9.93E + 02 | 7.44E − 05 | 1.29E − 06 |
| 8 | 418.879 | 13.246 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 |
| 9 | 471.239 | 14.902 | −2.36E − 04 | 6.00E + 02 | −2.72E − 05 | −3.66E − 07 |
| 10 | 523.599 | 16.558 | −9.73E + 02 | −4.25E − 04 | −4.32E − 07 | 3.56E − 05 |
| 11 | 575.959 | 18.213 | 1.93E − 04 | −4.02E + 02 | 1.22E − 05 | 1.34E − 07 |
| 12 | 628.319 | 19.869 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 |
| 13 | 680.678 | 21.525 | −1.64E − 04 | 2.88E + 02 | −6.22E − 06 | −5.80E − 08 |
| 14 | 733.038 | 23.181 | −4.96E + 02 | −3.04E − 04 | −8.00E − 08 | 9.25E − 06 |
| 15 | 785.398 | 24.836 | 1.42E − 04 | −2.16E + 02 | 3.51E − 06 | 2.83E − 08 |
| 16 | 837.758 | 26.492 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 |
| 17 | 890.118 | 28.148 | −1.25E − 04 | 1.68E + 02 | −2.13E − 06 | −1.51E − 08 |
| 18 | 942.478 | 29.804 | −3.00E + 02 | −2.36E − 04 | −2.27E − 08 | 3.38E−06 |
| 19 | 994.838 | 31.46 | 1.12E − 04 | −1.35E + 02 | 1.36E − 06 | 8.67E − 09 |
| 20 | 1047.198 | 33.115 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 |
