Modal

class sharpy.solvers.modal.Modal

Modal solver class, inherited from BaseSolver

Extracts the M, K and C matrices from the Fortran library for the beam. Depending on the choice of modal projection, these may or may not be transformed to a state-space form to compute the eigenvalues and mode shapes of the structure.

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

Name	Type	Description	Default
print_info	bool	Write status to screen	True
rigid_body_modes	bool	Write modes with rigid body mode shapes	False
use_undamped_modes	bool	Project the modes onto undamped mode shapes	True
NumLambda	int	Number of modes to retain	20
keep_linear_matrices	bool	Save M, C and K matrices to output dictionary	True
write_modes_vtk	bool	Write Paraview files with mode shapes	True
print_matrices	bool	Write M, C and K matrices to file	False
write_dat	bool	Write mode shapes, frequencies and damping to file	True
continuous_eigenvalues	bool	Use continuous time eigenvalues	False
dt	float	Time step to compute discrete time eigenvalues	0
delta_curved	float	Threshold for linear expressions in rotation formulas	0.01
plot_eigenvalues	bool	Plot to screen root locus diagram	False
max_rotation_deg	float	Scale mode shape to have specified maximum rotation	15.0
max_displacement	float	Scale mode shape to have specified maximum displacement	0.15
use_custom_timestep	int	If > -1, it will use that time step geometry for calculating the modes	-1
rigid_modes_cg	bool	Modify the ridid body modes such that they are defined wrt to the CG	False

free_free_modes(phi, M)

Warning

This function is deprecated. See free_modes_principal_axes() for a transformation to the CG and with respect to the principal axes of inertia.

Returns the rigid body modes defined with respect to the centre of gravity

The transformation from the modes defined at the FoR A origin, \(\boldsymbol{\Phi}\), to the modes defined using the centre of gravity as a reference is

\[\boldsymbol{\Phi}_{rr,CG}|_{TRA} = \boldsymbol{\Phi}_{RR}|_{TRA} + \tilde{\mathbf{r}}_{CG} \boldsymbol{\Phi}_{RR}|_{ROT}\]

\[\boldsymbol{\Phi}_{rr,CG}|_{ROT} = \boldsymbol{\Phi}_{RR}|_{ROT}\]

Returns

Transformed eigenvectors

Return type

(np.array)

run()

Extracts the eigenvalues and eigenvectors of the clamped structure.

If use_undamped_modes == True then the free vibration modes of the clamped structure are found solving:

\[\mathbf{M\,\ddot{\eta}} + \mathbf{K\,\eta} = 0\]

that flows down to solving the non-trivial solutions to:

\[(-\omega_n^2\,\mathbf{M} + \mathbf{K})\mathbf{\Phi} = 0\]

On the other hand, if the damped modes are chosen because the system has damping, the free vibration modes are found solving the equation of motion of the form:

\[\mathbf{M\,\ddot{\eta}} + \mathbf{C\,\dot{\eta}} + \mathbf{K\,\eta} = 0\]

which can be written in state space form, with the state vector \(\mathbf{x} = [\eta^T,\,\dot{\eta}^T]^T\) as

\[\begin{split}\mathbf{\dot{x}} = \begin{bmatrix} 0 & \mathbf{I} \\ -\mathbf{M^{-1}K} & -\mathbf{M^{-1}C} \end{bmatrix} \mathbf{x}\end{split}\]

and therefore the mode shapes and frequencies correspond to the solution of the eigenvalue problem

\[\mathbf{A\,\Phi} = \mathbf{\Lambda\,\Phi}.\]

From the eigenvalues, the following system characteristics are provided:

• Natural Frequency: \(\omega_n = |\lambda|\)

• Damped natural frequency: \(\omega_d = \text{Im}(\lambda) = \omega_n \sqrt{1-\zeta^2}\)

• Damping ratio: \(\zeta = -\frac{\text{Re}(\lambda)}{\omega_n}\)

In addition to the above, the modal output dictionary includes the following:

• M: Tangent mass matrix

• C: Tangent damping matrix

• K: Tangent stiffness matrix

• Ccut: Modal damping matrix \(\mathbf{C}_m = \mathbf{\Phi}^T\mathbf{C}\mathbf{\Phi}\)

• Kin_damp: Forces gain matrix (when damped): \(K_{in} = \mathbf{\Phi}_L^T \mathbf{M}^{-1}\)

• eigenvectors: Right eigenvectors

• eigenvectors_left: Left eigenvectors given when the system is damped

Returns

updated data object with modal analysis as part of the last structural time step.

Return type

PreSharpy