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 available.

The beam state space has information on the states which will depend on whether the system is modal or expressed in physical coordinates.

If modal the state variables will be q and q_dot representing the modal displacements and the time derivatives.

If nodal and free-flying, the state variables will be eta for the flexible degrees of freedom (displacements and CRVs for each node (dim6)), V representing the linear velocities at the A frame (dim3), W representing the angular velocities at the A frame (dim3), and orient representing the orientation variable of the A frame with respect to G

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	True
print_info	bool	Display information on screen	True
gravity	bool	Linearise gravitational forces	False
remove_dofs	list(str)	Remove desired degrees of freedom (flexible DOFs, linear velocities, rotational velocities, orientation)	[]	eta, V, W, orient
remove_sym_modes	bool	Remove symmetric modes if wing is clamped	False
remove_rigid_states	bool	(For Stability Derivatives) - Remove RIGID STATES from SS leaving input/output channels unchanged	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:

recover_accelerations(full_ss)

For a system with displacement and velocity outputs (full_ss), recover the accelerations and append them as new output channels.

This function produces an output gain that should then be connected in series to the desired system

Parameters:

full_ss (libss.StateSpace) – State space for which to provide output gain to recover accelerations

Returns:

Gain adding the accelerations as new output channels

Return type:

libss.Gain

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.

trim_nodes(trim_list=<class 'list'>)

Removes degrees of freedom from the second order system.

Parameters:

trim_list (list) – List of degrees of freedom to remove eta, V, W or orient

unpack_flex_dof(eta, eta_dot=None)

Unpacks a vector of structural displacements and velocities into a SHARPy familiar form of pos, psi and their time derivatives

Parameters:

• eta (np.array) – Vector of structural displacements

• eta_dot (np.array (Optional) – Vector of structural velocities

Returns:

Containing pos, psi, pos_dot, psi_dot if eta_dot is provided, else

only the displacements are returned

Return type:

tuple

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