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Spectral response analysis |
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Spectral response analysis |
The spectral response analysis capability of OrcaFlex allows you to determine the response characteristics for any OrcaFlex result. This feature produces output similar to a frequency domain solve but the calculation is based on a random wave time domain simulation. The results of this simulation are transformed into the frequency domain using a fast Fourier transform (FFT) and the spectral response is then calculated. The final output of the analysis is the response amplitude operator (RAO) for the result of interest.
In order to calculate spectral response you must first perform a random-wave time-domain simulation of the system of interest. To do this you must specify a single wave train with wave type of response calculation. This random wave type has a truncated white noise spectrum with the energy spread evenly over a user-defined range of frequencies.
The spectral response analysis starts from the time history of the result of interest. The time history covers the response calculation simulation period given on the general data form. The duration of this period, together with the logging interval $\delta t$, determine the total number of samples available for the FFT.
OrcaFlex does not necessarily use all the available samples. This is because the FFT calculation is slow when presented with a time history whose size is a large prime or a product of large primes. Suppose that $N$ is the total number of available samples. OrcaFlex will choose a value $M$ with the properties that (i) $M{\leq}N$ and (ii) it is known that the FFT can be calculated quickly and efficiently for $M$ samples. Having chosen $M$, OrcaFlex selects the $M$ samples from the original time history that are closest to the end of the simulation.
OrcaFlex then calculates the power spectral density (PSD) of these $M$ time history samples using the FFT. The PSD is denoted as the sequence $\{f_i,P_i\}$ for $i=1,2,\ldots,\frac{M}{2}$, where $P_i$ is understood to be the PSD for frequency $f_i$. The values $f_i$ are integer multiples of the FFT's fundamental frequency, $\Delta f_i{=}\frac1{\delta tM}$, given by $f_i{=}i\Delta f$. The maximum frequency, $\frac{M}{2}\Delta f = \frac{1}{2\delta t}$, is known as the Nyquist critical frequency.
Finally, the RAO is calculated as \begin{equation} R_i = \sqrt{\frac{P_i}{S_i}} \text{ for }i=1,2,\ldots,\frac{M}{2} \end{equation} where $R_i$ is the RAO at frequency $f_i$ and $S_i$ is the spectral density at frequency $f_i$ for the response calculation random wave.
| Notes: | Provided that a response calculation wave type has been selected the waves page of the environment data form reports the value of the number of data points, $M$, which will be used. |
| Only the frequencies which lie in the target frequency range are used for wave components. Consequently, there may be fewer than $\frac{M}{2}$ wave components. |
It is important that the random wave components have frequencies which match those produced by the FFT. This is because of a phenomenon of the FFT known as frequency leakage which would occur if the random wave component frequencies did not match the FFT frequencies. The effect of leakage is to make the output of the FFT noisy.
The response calculation wave components are selected with frequencies that are also integer multiples of $\Delta f$. In this way the frequency leakage effect is avoided.
Not all these frequencies are used in the response calculation random wave. This is because there could potentially be so many frequencies (i.e. for large values of $M/2$) that the real-time required to simulate a wave with that many components would be prohibitive. You control the range of frequencies to be used with the target frequency range data on the environment data form.
For frequency domain approaches to calculating system responses each nonlinearity in the system has to be handled in special ways. However, by calculating the RAOs using a fully nonlinear time domain simulation and then transforming to the frequency domain using Fourier transform methods, the nonlinearities are included in the calculation automatically.
The advantage of the method used by OrcaFlex is that nonlinearities can be handled implicitly without the need for special, bespoke linearisation techniques.
| Warning: | The generated response RAOs are not always accurate. Depending on the type of system, it may be impossible to calculate good quality response RAOs using this method. You must use the output with caution. |