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
folder	str	Output folder	./output
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)

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