LinearBeam

class sharpy.linear.assembler.linearbeam.LinearBeam

State space member

Define class for linear state-space realisation of GEBM flexible-body equations from SHARPy timestep_info class and with the nonlinear structural information.

State-space models can be defined in continuous or discrete time (dt required). Modal projection, either on the damped or undamped modal shapes, is also avaiable.

Notes on the settings:

1. modal_projection={True,False}: determines whether to project the states

   onto modal coordinates. Projection over damped or undamped modal shapes can be obtained selecting:

   • proj_modes = {'damped','undamped'}

   while

   • inout_coords={'modes','nodal'}

   determines whether the modal state-space inputs/outputs are modal coords or nodal degrees-of-freedom. If modes is selected, the Kin and Kout gain matrices are generated to transform nodal to modal dofs

2. dlti={True,False}: if true, generates discrete-time system.

   The continuous to discrete transformation method is determined by:

   discr_method={ 'newmark',  # Newmark-beta
                       'zoh',              # Zero-order hold
                       'bilinear'} # Bilinear (Tustin) transformation
   

   DLTIs can be obtained directly using the Newmark-\(\beta\) method

   discr_method='newmark' newmark_damp=xx with xx<<1.0

   for full-states descriptions (modal_projection=False) and modal projection over the undamped structural modes (modal_projection=True and proj_modes). The Zero-order holder and bilinear methods, instead, work in all descriptions, but require the continuous state-space equations.

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

Name	Type	Description	Default	Options
modal_projection	bool	Use modal projection	True
inout_coords	str	Beam state space input/output coordinates	nodes	nodes, modes
num_modes	int	Number of modes to retain	10
discrete_time	bool	Assemble beam in discrete time	True
dt	float	Discrete time system integration time step	0.001
proj_modes	str	Use undamped or damped modes	undamped	damped, undamped
discr_method	str	Discrete time assembly system method:	newmark	newmark, zoh, bilinear
newmark_damp	float	Newmark damping value. For systems assembled using newmark	0.0001
use_euler	bool	Use euler angles for rigid body parametrisation	False
print_info	bool	Display information on screen	True
gravity	bool	Linearise gravitational forces	False
remove_dofs	list(str)	Remove desired degrees of freedom	[]	eta, V, W, orient
remove_sym_modes	bool	Remove symmetric modes if wing is clamped	False

assemble(t_ref=None)

Assemble the beam state-space system.

Parameters

t_ref (float) – Scaling factor to non-dimensionalise the beam’s time step.

Returns:

remove_symmetric_modes()

Removes symmetric modes when the wing is clamped at the midpoint.

It will force the wing tip displacements in z to be postive for all modes.

Updates the mode shapes matrix, the natural frequencies and the number of modes.

unpack_ss_vector(x_n, u_n, struct_tstep)

Warning

Under development. Missing:

• Accelerations

• Double check the cartesian rotation vector

• Tangential operator for the moments

Takes the state \(x = [\eta, \dot{\eta}]\) and input vectors \(u = N\) of a linearised beam and returns a SHARPy timestep instance, including the reference values.

Parameters

• x_n (np.ndarray) – Structural beam state vector in nodal space

• y_n (np.ndarray) – Beam input vector (nodal forces)

• struct_tstep (utils.datastructures.StructTimeStepInfo) – Reference timestep used for linearisation

Returns

new timestep with linearised values added to the reference value

Return type

utils.datastructures.StructTimeStepInfo