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Frequency domain analysis theory

Frequency domain simulations can be used to quickly analyze the dynamic response of lines provided that the system does not exhibit high non- linearities.

Mechanical equations

The general equation of motion can be written (for the translation degrees of freedom): $$ KX+B\dot X+M\ddot X=F~~Eq.1 $$

Where

• \(K\) is the stiffness matrix,

• \(B\) is the damping matrix,

• \(M\) is the mass matrix,

• \(F\) is the loading,

• \(X\) is the value of the translation with its time derivatives noted by a dot.

For the frequency domain analysis, the solution \(X(t)\) is written as the sum of a static component and a frequency component. The latter is itself divided into imposed motion response (in order to satisfy the cinematic boundary conditions) and a component which is solution of the mechanical system:

\[ X(t)=X_{stat}+\displaystyle\sum_{i=1}^{Nbfreq}\left\{ Re\left\{ \Bigr[ \sum_{imp=1}^{N_{imp}} a_{imp}(\omega_i)X_{imp}+X_f (\omega_i) \Bigr] e^{-j\omega_i t} \right\} \right\}~~Eq.2 \]

In Eq. 2, the static part is obtained by computing the static equilibrium: $$ K_{stat}X_{stat}=F_{stat}~~Eq.3 $$

For the imposed motion, aimp is the complex coefficient associated with the motion and \(x_{imp}\) is obtained by solving: $$ K_{stat}X_{imp}=0~~Eq.4 $$

In effect, the stiffness matrix is the sum of a static and a frequency dependent matrix. The damping and mass matrices are also decomposed into two matrices: one frequency independent, one frequency dependent. The former can be computed one and for all, at the beginning of the frequency calculations.

Finally, the dynamic loading are considered to be expressed as a complex coefficient multiplied by \(e^{jwt}\), and the mechanical equations (Eq. 1) becomes: $$ [K(\omega)-j\omega(B+B_a(\omega))-\omega^2(M+M_a(\omega))]X(\omega)=F(\omega)~~Eq.5 $$

Linearization

The hydrodynamic force in mooring lines and risers is expressed by the Morison formula. Specifically, the drag force is proportional to the product of the relative velocity of the fluid with respect to the structure multiplied by its norm. Therefore, it is necessary to introduce a linearization as indicated in Eq. 6.

\[ ||v_{relative}||v_{relative}\approx\Omega(v_{relative})v_{relative}~~Eq.6 \]

The linearization coefficients are divided in two times two cases :

• Without and with current,

• In regular or irregular waves.

The four cases are implemented in DeepLines. Note that the case with current introduces an additional static force. At the present time, this extra static force in not taken into account. It is considered that the static equilibrium is not dependant on the frequency.

General algorithm

The methodology requires to perform first a static calculation to obtain the equilibrium, then imposed displacements are derived. Finally, the linear frequency domain system is built and solved. Due to the drag formulation, the linearization coefficients are function of the solution. Therefore an iteration technique is used to obtain the converge solution. The algorithm can be summarized by the following steps:

1. Calculation of the static equilibrium

2. Calculation of the imposed motion coefficients

3. Calculation of the frequency domain response

   a. Building of the frequency independent matrices

   b. For each frequency, the frequency dependent matrices are obtained, as well as the loading and the linear system is solved.

There are two different approaches depending on the type of waves used in the simulation. For regular wave, the normalization coefficient is dependent on the single wave frequency, while for irregular waves, the coefficients are dependent on all the components of the waves representing the wave spectrum. Therefore, for regular waves, calculations are performed independently for each period. For irregular waves, calculations are performed on all frequencies, then the linearization coefficient is recomputed and the next round of calculations on all periods is performed.