The Frequency Response Method

The static and dynamic response analysis is based on a three dimensional (3D) finite element formulation that in principle may take any 3D effects into account. The element theory is described in the RIFLEX Theory Description.

The beam element accounts for the geometric stiffness from the effective tension in the structure, and can handle large displacements and rotations in three dimensions. Non-linear material and geometric effects can also be accounted for. The dynamic analysis in VIVANA applies the frequency response method and is hence linear with respect to structural stiffness. This limitation will normally not be a problem since VIV amplitudes are small relative to the global dimensions of the structure, which means that the structural behaviour is linear.

Vortex induced vibration is assumed to take place at discrete frequencies. Laboratory tests and large-scale measurements support this assumption. A certain bandwidth of the response, in particular at high velocities (referred to the lowest possible velocity that will initiate VIV), may be present, but the discrete frequency assumption is still applied in VIVANA.

The frequency response method is convenient for calculation of VIV, in particular because equilibrium iteration to account for response dependent forces is easy to formulate. A Newton-Raphson scheme is applied. For a converged solution we have consistency between the response amplitude and the excitation force coefficient at all nodes along the structure, and there is also consistency between the phase of the load and the phase of the response. The phase consistency is needed since the excitation force is defined as the component of the hydrodynamic force that is in phase with the velocity. The two criteria must hence be satisfied at all nodes within the excitation zone for the actual frequency.

The frequency response method is able to describe both standing and travelling waves. Normal modes are not referred to in this method, but the mode shape for the actual response frequency serves as a starting point for the iteration process. The solution may, however, be considered to be a complex mode since it consists of a complex vector that is equivalent to the solution of free oscillations for a damped system. Hence, solving the frequency response equation is mathematically equivalent to solving the complex eigenvalue equation with some negative and some positive damping terms. Numerically, however, the two methods are different, which in fact is the reason why the frequency response method was selected for this application.

In the case of discrete frequencies the response may be calculated by using finite elements and the frequency response method. The equation of dynamic equilibrium is normally written

The external loads \(\mathrm {\boldsymbol{R}}\) will in our case be harmonic, but loads at all degrees of freedom are not necessarily in phase. It is convenient to describe this type of load by a complex load vector with harmonic time variation

where \(\mathrm {\boldsymbol{X}}\) is a complex vector and \(\mathrm {\omega }\) is the load frequency. Note that the phase for the load at a specific degree of freedom is defined by the relative magnitude of the real and imaginary components of the associated element in the complex load vector.

The response vector \(\mathrm {\boldsymbol{r}}\) will also be given by a complex vector \(\mathrm {\boldsymbol{x}}\) and a harmonic time variation. Hence we have

The phase for the response at each degree of freedom is defined by the complex response vector in the same way as for the load.

By introducing the hydrodynamic mass and damping matrices, dynamic equilibrium can now be expressed as:

where

In this equation, it is necessary to define the hydrodynamic damping matrix, and also the load vector as functions of the response vector. Iteration is hence needed when solving this equation. Two iteration procedures are available; one based on direct use of intermediate response results for updating of excitation coefficients, and the other based on the classical Newton-Raphson method that requires calculation of gradients for the unbalanced forces. Note that the response frequency is fixed during this iteration; cf. Step 3 in the analysis procedure, given in Method Overview. The iteration will identify response shape and amplitude that give consistency between the response level and applied hydrodynamic coefficients at all translation degrees of freedom.

The solution of this equation can formally be written

The matrix \(\mathrm {\boldsymbol{H(\omega )}}\) is referred to as the frequency response or transfer function matrix. Each element \(\mathrm {h_{ij}}\) in the matrix defines the response at degree of freedom \(\mathrm {i}\) from a harmonic unit load with frequency \(\mathrm {\omega }\) at degree of freedom \(\mathrm {j}\). Hence, the \(\mathrm {\boldsymbol{H}}\) matrix can be understood as a discrete Greens function for the structure.