# LinearBeam¶

class sharpy.linear.assembler.linearbeam.LinearBeam[source]

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)[source]

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)[source]

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()[source]

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'>)[source]

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)[source]

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)[source]

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