OrcaFlex Manual - Orcina

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Modal Analysis, Theory
If the system hangs in one of the global axis planes then you can often distinguish whether a mode is in-plane or outof-plane
by looking at the pattern of zeros in the table of displacements. For example if the system hangs in the XZ
plane then the out-of-plane modes have non-zero Y-displacements but zero (or very small) X- and Z-displacements,
and the in-plane modes have the opposite pattern of zeros.
Outline Theory
Modal analysis is a standard technique that is well-documented in the literature, but here is a brief outline. First
consider a single degree of freedom system consisting of a mass attached to a linear spring. The undamped equation
of motion is:
Mx''(t) = -Kx(t)
where x(t) is the offset (at time t) from mean position, x''(t) is the acceleration, M is its mass and K is the stiffness of
the spring. Since this analysis neglects any damping the results are referred to as the undamped modes.
The solution of the equation is known to be simple harmonic, i.e. of the form x(t) = a.sin(ωt), where a and ω are
unknowns to be found by solving the equation. Differentiating x(t) gives:
x''(t) = -ω 2 .a.sin(ωt)
so when we substitute into the equation of motion we obtain:
-M.ω 2 .a.sin(ωt) = -K.a.sin(ωt) (1)
which can be rearranged to give:
ω = (K/M) ½ .
This is the angular frequency of the oscillation and so the natural period T is given by:
T = 2π(M/K) ½
For this simple harmonic oscillator there is just a single undamped natural mode, corresponding to the single degree
of freedom. For a continuous riser there are an infinite number of degrees of freedom, and hence an infinite number
of undamped natural modes, but computers work with discretised models with finite numbers of degrees of
freedom.
Consider a discretised line in OrcaFlex with N degrees of freedom. In this situation the above equations still apply,
but they now have to be interpreted as matrix/vector equations where ω and T remain scalars, a, x and x'' become
vectors with N elements, and M and K become N×N matrices.
Equation (1) is an eigen-problem with N solutions, the i th solution being ωi and ai, say, where ωi is a scalar and ai is a
vector with N elements. This i th solution is called the i th natural mode. It is an oscillation of the line in which all the
degrees of freedom oscillate at the same angular frequency ωi. But different degrees of freedom have different
amplitudes, given by the components of ai. This amplitude variation is called the mode's shape.
Eigen-solvers
Two eigen-solvers are used to perform modal analysis. The choice of which to use is made based on the number of
modes extracted, n, and the number of degrees of freedom, N.
If n ≤ N/3 and n ≤ 1000 then an iterative Lanczos algorithm will be used. Otherwise a direct method based on
tridiagonal MATRIX diagonalisation is used. For large problems the iterative Lanczos algorithm is much faster and
requires much less memory and so should be used if at all possible.
One final subtlety concerns the precise definition of n in the above inequalities. The Lanczos algorithm works by
finding the largest (or smallest) eigenvalue first, then the next largest (or smallest) and so on. Consequently if you
ask for modes 5 to 10 then the solver has to find modes 1 to 4 first and so the number of modes extracted, n, is 10.
Seabed friction
The theory outlined above requires that the mass and stiffness matrices are symmetric which is not always the case
in an OrcaFlex model. The most important example of this is the friction model. Friction is a non-conservative effect
and non-conservatism equates to non-symmetric terms in the stiffness matrix. Clearly this presents a problem.
The non-conservatism of the standard OrcaFlex friction model arises when a node is slipping, that is when the
deflection from its friction target position exceeds Dcrit. When performing modal analysis OrcaFlex assumes that
nodes on the seabed are restrained by a linear stiffness effect determined by the seabed's shear stiffness, Ks and the
node's contact area, A. This stiffness term corresponds to the stiffness of a linear spring acting in the plane of the
seabed, connecting the node and its target position, and with a stiffness of KsA.