AsymptoticStability

class sharpy.postproc.asymptoticstability.AsymptoticStability

Calculates the asymptotic stability properties of the linearised system by computing the corresponding eigenvalues and eigenvectors.

The output of this solver is written to the stability directory in the case output.

The stability of the systems specified in target_systems is performed. If the system has been previously scaled, a reference_velocity should be provided to compute the stability at such point.

The eigenvalues can be truncated, keeping a minimum num_evals (sorted by decreasing real part) or by limiting the higher frequency modes through frequency_cutoff.

Results can be saved to file using export_eigenvalues. The setting display_root_locus shows a simple Argand diagram where the continuous time eigenvalues are displayed.

The eigenvectors can be displayed (in .vtu format for use in Paraview) with the modes_to_plot setting, whereby the user specifies the mode indices that are to be plotted (sorted with in the same way as the eigenvalue table, i.e. by decreasing real part). A snapshot of the eigenvector is produced for the 0, 45, 90 and 135 degree phases. This feature currently supports the flexible structural modes and does not show the rigid body contribution. The output is written to the stability/modes folder and includes the structure at the reference linearisation state.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
print_info	bool	Print information and table of eigenvalues	False
reference_velocity	float	Reference velocity at which to compute eigenvalues for scaled systems	1.0
frequency_cutoff	float	Truncate higher frequency modes. If zero none are truncated	0
export_eigenvalues	bool	Save eigenvalues and eigenvectors to file.	False
output_file_format	str	Eigenvalue/eigenvector output file format. HDF5 or text (.dat) files.	dat
display_root_locus	bool	Show plot with eigenvalues on Argand diagram	False
velocity_analysis	list(float)	List containing min, max and number of velocities to analyse the system	[]
target_system	list(str)	System or systems for which to find frequency response.	['aeroelastic']
num_evals	int	Number of eigenvalues to retain.	200
modes_to_plot	list(int)	List of mode numbers to plot. Plots the 0, 45, 90 and 135degree phases.	[]

compute_eigenvalues(ss, system_name_list=None, not_scaled=True)

Computes the eigenvalues and eigenvectors of the state-space

Parameters:

• ss (libss.StateSpace or list([libss.StateSpace]) – State-space or list of state-spaces

• system_name_list (list([str]) – Names of systems in the case multiple systems are required

• not_scaled (bool) – Flag to indicate whether the systems are assembled in non-dimensional time

convert_to_continuoustime(dt, discrete_time_eigenvalues, not_scaled=False)

Convert eigenvalues to discrete time. The not_scaled argument can be used to bypass the search from within SHARPy of scaling factors. For instance, when the state-space of choice is not part of a standard SHARPy case but rather an interpolated ROM etc.

The eigenvalues are converted to continuous time using

\[\lambda_{ct} = \frac{\log (\lambda_{dt})}{\Delta t}\]

If the system is scaled, the dimensional time step is retrieved as

\[\Delta t_{dim} = \bar{\Delta t} \frac{l_{ref}}{U_{\infty, actual}}\]

where \(l_{ref}\) is the reference length and \(U_{\infty, actual}\) is the free stream velocity at which to calculate the eigenvalues.

Parameters:

• dt (float) – Discrete time increment.

• discrete_time_eigenvalues (np.ndarray) – Array of discrete time eigenvalues.

• not_scaled (bool) – Treat the system as not scaled. No Scaling Factors will be searched in SHARPy.

static display_root_locus(eigenvalues)

Displays root locus diagrams.

Returns the fig and ax handles for further editing.

Returns:

ax:

Return type:

fig

export_eigenvalues(num_evals, eigenvalues, eigenvectors, filename=None)

Saves a num_evals number of eigenvalues and eigenvectors to file.

The files are saved in the output directory and include:

• {system_name}_eigenvalues.dat: Array of eigenvalues of shape (num_evals, 2) where the first column corresponds to the real part and the second column to the imaginary part.

• {system_name}_stability.h5: An .h5 file containing the desired number of eigenvalues and eigenvectors of the chosen systems.

The units of the eigenvalues are rad/s.

Parameters:

• num_evals (int) – Number of eigenvalues to save.

• eigenvalues (np.ndarray) – Eigenvalue array.

• eigenvectors (np.ndarray) – Matrix of eigenvectors.

• filename (str (optional)) – Optional prefix of the output filenames.

See also

Loading and saving complex arrays: https://stackoverflow.com/questions/6494102/how-to-save-and-load-an-array-of-complex-numbers-using-numpy-savetxt/6522396

plot_modes(eigenvectors)

Plot the aeroelastic mode shapes for the first n_modes_to_plot

Plots the 0, 45, 90 and 135 degrees phase of the mode. Also plots the reference at the linearisation state.

\[x_{out} = Re(\Phi_i e^{i\theta})\]

for \(\theta \in \{0, \pi/4, \pi/2, 3\pi/4}\).

run(**kwargs)

Computes the eigenvalues and eigenvectors

Returns:

Eigenvalues sorted and frequency truncated eigenvectors (np.ndarray): Corresponding mode shapes

Return type:

eigenvalues (np.ndarray)

static sort_eigenvalues(eigenvalues, eigenvectors, frequency_cutoff=0, number_of_eigenvalues=None)

Sort continuous-time eigenvalues by order of magnitude.

The conjugate of complex eigenvalues is removed, then if specified, high frequency modes are truncated. Finally, the eigenvalues are sorted by largest to smallest real part.

Parameters:

• eigenvalues (np.ndarray) – Continuous-time eigenvalues

• eigenvectors (np.ndarray) – Corresponding right eigenvectors

• frequency_cutoff (float) – Cutoff frequency for truncation [rad/s]

• number_of_eigenvalues (int (optional)) – Number of eigenvalues to retain

Returns:

eigenvalues and eigenvectors

Return type:

tuple(np.array, np.array)

velocity_analysis()

Velocity analysis for scaled systems.

Runs the stability analysis for different velocities for aeroelastic systems that have been previously scaled.

For every velocity, the linear system is updated. This involves updating the structural matrices and the coupling matrix. The eigenvalues saved are in continuous time.

It saves the results to a .dat file where the first column corresponds to the free stream velocity and the second and third columns to the real and imaginary parts of the eigenvalues.