Source code for sharpy.linear.assembler.linearaeroelastic

import numpy as np
import scipy.linalg as sclalg
import warnings

import sharpy.linear.utils.ss_interface as ss_interface
import sharpy.linear.src.libss as libss
import sharpy.utils.settings as settings
import sharpy.utils.cout_utils as cout
import sharpy.utils.algebra as algebra
import sharpy.utils.generator_interface as gi
from sharpy.linear.utils.ss_interface import LinearVector, InputVariable, StateVariable, OutputVariable
import sharpy.aero.utils.utils as aero_utils


[docs]@ss_interface.linear_system class LinearAeroelastic(ss_interface.BaseElement): r""" Assemble a linearised aeroelastic system The aeroelastic system can be seen as the coupling between a linearised aerodynamic system (System 1) and a linearised beam system (System 2). The coupled system retains inputs and outputs from both systems such that .. math:: \mathbf{u} = [\mathbf{u}_1;\, \mathbf{u}_2] and the outputs are also ordered in a similar fashion .. math:: \mathbf{y} = [\mathbf{y}_1;\, \mathbf{y}_2] Reference the individual systems for the particular ordering of the respective input and output variables. """ sys_id = 'LinearAeroelastic' settings_default = dict() settings_types = dict() settings_description = dict() settings_types['aero_settings'] = 'dict' settings_default['aero_settings'] = None settings_description['aero_settings'] = 'Linear UVLM settings' settings_types['beam_settings'] = 'dict' settings_default['beam_settings'] = None settings_description['beam_settings'] = 'Linear Beam settings' settings_types['uvlm_filename'] = 'str' settings_default['uvlm_filename'] = '' settings_description['uvlm_filename'] = 'Path to .data.h5 file containing UVLM/ROM state space to load' settings_types['track_body'] = 'bool' settings_default['track_body'] = True settings_description['track_body'] = 'UVLM inputs and outputs projected to coincide with lattice at linearisation' settings_types['use_euler'] = 'bool' settings_default['use_euler'] = True settings_description['use_euler'] = 'Parametrise orientations in terms of Euler angles' settings_table = settings.SettingsTable() __doc__ += settings_table.generate(settings_types, settings_default, settings_description) def __init__(self): self.ss = None # The state space object self.lsys = dict() # Contains the underlying objects self.uvlm = None self.beam = None self.load_uvlm_from_file = False self.settings = dict() self.state_variables = None self.couplings = dict() self.linearisation_vectors = dict() # Aeroelastic coupling gains # transfer self.Kdisp = None self.Kvel_disp = None self.Kdisp_vel = None self.Kvel_vel = None self.Kforces = None # stiffening factors self.Kss = None self.Krs = None self.Csr = None self.Crs = None self.Crr = None def initialise(self, data): try: self.settings = data.settings['LinearAssembler']['linear_system_settings'] except KeyError: self.settings = None settings.to_custom_types(self.settings, self.settings_types, self.settings_default, no_ctype=True) if self.settings['use_euler']: self.settings['beam_settings']['use_euler'] = True # Create Linear UVLM self.uvlm = ss_interface.initialise_system('LinearUVLM') self.uvlm.initialise(data, custom_settings=self.settings['aero_settings']) # Get the minimum parameters needed to define the wake vel_gen_name, vel_gen_settings = aero_utils.find_velocity_generator(data.settings) vel_gen_type = gi.generator_from_string(vel_gen_name) vel_gen = vel_gen_type() vel_gen.initialise(vel_gen_settings) wake_prop_settings = {'dt': self.settings['aero_settings']['dt'], 'ts': data.ts, 't': data.ts*self.settings['aero_settings']['dt'], 'for_pos': data.structure.timestep_info[-1].for_pos, 'cfl1': self.settings['aero_settings']['cfl1'], 'vel_gen': vel_gen} if self.settings['uvlm_filename'] == '': self.uvlm.assemble(track_body=self.settings['track_body'], wake_prop_settings=wake_prop_settings) else: self.load_uvlm_from_file = True # Create beam self.beam = ss_interface.initialise_system('LinearBeam') self.beam.initialise(data, custom_settings=self.settings['beam_settings']) for k, v in self.uvlm.linearisation_vectors.items(): self.linearisation_vectors[k] = v for k, v in self.beam.linearisation_vectors.items(): self.linearisation_vectors[k] = v self.get_gebm2uvlm_gains(data)
[docs] def assemble(self): r""" Assembly of the linearised aeroelastic system. The UVLM state-space system has already been assembled. Prior to assembling the beam's first order state-space, the damping and stiffness matrices have to be modified to include the damping and stiffenning terms that arise from the linearisation of the aeordynamic forces with respect to the A frame of reference. See :func:`sharpy.linear.src.lin_aeroela.get_gebm2uvlm_gains()` for details on the linearisation. Then the beam is assembled as per the given settings in normalised time if the aerodynamic system has been scaled. The discrete time systems of the UVLM and the beam must have the same time step. The UVLM inputs and outputs are then projected onto the structural degrees of freedom (obviously with the exception of external gusts and control surfaces). Hence, the gains :math:`\mathbf{K}_{sa}` and :math:`\mathbf{K}_{as}` are added to the output and input of the UVLM system, respectively. These gains perform the following relation: .. math:: \begin{bmatrix}\zeta \\ \zeta' \\ u_g \\ \delta \end{bmatrix} = \mathbf{K}_{as} \begin{bmatrix} \eta \\ \eta' \\ u_g \\ \delta \end{bmatrix} = .. math:: \mathbf{N}_{nodes} = \mathbf{K}_{sa} \mathbf{f}_{vertices} If the beam is expressed in modal form, the UVLM is further projected onto the beam's modes to have the following input/output structure: Returns: """ uvlm = self.uvlm beam = self.beam # Linearisation of the aerodynamic forces introduces stiffenning and damping terms into the beam matrices flex_nodes = self.beam.sys.num_dof_flex rigid_dof = beam.sys.Kstr.shape[0] - flex_nodes total_dof = flex_nodes + rigid_dof if uvlm.scaled: beam.assemble(t_ref=uvlm.sys.ScalingFacts['time']) else: beam.assemble() if not self.load_uvlm_from_file: # Projecting the UVLM inputs and outputs onto the structural degrees of freedom Ksa = self.Kforces[:beam.sys.num_dof, :] # maps aerodynamic grid forces to nodal forces gain_ksa = libss.Gain(Ksa) gain_ksa.input_variables = LinearVector.transform(uvlm.ss.output_variables, to_type=InputVariable) if beam.sys.Kin is not None: gain_ksa.output_variables = LinearVector.transform(beam.sys.Kin.input_variables, to_type=OutputVariable) else: gain_ksa.output_variables = LinearVector.transform(beam.ss.input_variables, to_type=OutputVariable) # Map the nodal displacement and velocities onto the grid displacements and velocities Kas = np.zeros((uvlm.ss.inputs, 2*beam.sys.num_dof + (uvlm.ss.inputs - 2*self.Kdisp.shape[0]))) Kas[:2*self.Kdisp.shape[0], :2*beam.sys.num_dof] = \ np.block([[self.Kdisp[:, :beam.sys.num_dof], self.Kdisp_vel[:, :beam.sys.num_dof]], [self.Kvel_disp[:, :beam.sys.num_dof], self.Kvel_vel[:, :beam.sys.num_dof]]]) # Retain other inputs Kas[2*self.Kdisp.shape[0]:, 2*beam.sys.num_dof:] = np.eye(uvlm.ss.inputs - 2 * self.Kdisp.shape[0]) gain_kas = libss.Gain(Kas) gain_kas.output_variables = LinearVector.transform(uvlm.ss.input_variables, to_type=OutputVariable) if beam.sys.Kout is not None: kas_in_vars = LinearVector.transform(beam.sys.Kout.output_variables, to_type=InputVariable) else: kas_in_vars = LinearVector.transform(beam.ss.output_variables, to_type=InputVariable) for variable in uvlm.ss.input_variables: if variable.name not in ['zeta', 'zeta_dot']: kas_in_vars.append(variable) gain_kas.input_variables = kas_in_vars # Scaling if uvlm.scaled: Kas /= uvlm.sys.ScalingFacts['length'] uvlm.connect_output(gain_ksa) uvlm.connect_input(gain_kas) # Stiffenning and damping terms within the uvlm Dmod = np.zeros_like(uvlm.ss.D) Dmod[:flex_nodes, :flex_nodes] -= self.Kss if rigid_dof > 0: Dmod[flex_nodes:, :flex_nodes] -= self.Krs Dmod[flex_nodes:, total_dof: total_dof + flex_nodes] -= self.Crs Dmod[:flex_nodes, total_dof + flex_nodes: 2 * total_dof] -= self.Csr Dmod[flex_nodes:, total_dof + flex_nodes: 2 * total_dof] -= self.Crr if uvlm.scaled: Dmod /= uvlm.sys.ScalingFacts['force'] uvlm.ss.D += Dmod self.couplings['Ksa'] = gain_ksa self.couplings['Kas'] = gain_kas if self.settings['beam_settings']['modal_projection'] is True and \ self.settings['beam_settings']['inout_coords'] == 'modes': # Project UVLM onto modal space phi = beam.sys.U in_mode_matrix = np.zeros((uvlm.ss.inputs, beam.ss.outputs + (uvlm.ss.inputs - 2*beam.sys.num_dof))) in_mode_matrix[:2*beam.sys.num_dof, :2*beam.sys.num_modes] = sclalg.block_diag(phi, phi) in_mode_matrix[2*beam.sys.num_dof:, 2*beam.sys.num_modes:] = np.eye(uvlm.ss.inputs - 2*beam.sys.num_dof) in_mode_gain = libss.Gain(in_mode_matrix) in_mode_inputs = LinearVector.transform(beam.ss.output_variables, to_type=InputVariable) LinearVector.check_same_vectors(in_mode_inputs, beam.ss.output_variables) for variable in uvlm.ss.input_variables.copy(): if variable.name not in ['eta', 'eta_dot', 'beta_bar', 'beta']: in_mode_inputs.append(variable) in_mode_gain.input_variables = in_mode_inputs in_mode_gain.output_variables = LinearVector.transform(uvlm.ss.input_variables, to_type=OutputVariable) out_mode_matrix = phi.T out_mode_gain = libss.Gain(out_mode_matrix, input_vars=LinearVector.transform(uvlm.ss.output_variables, to_type=InputVariable), output_vars=LinearVector.transform(beam.ss.input_variables, to_type=OutputVariable)) uvlm.connect_input(in_mode_gain) uvlm.connect_output(out_mode_gain) self.couplings['in_mode_gain'] = in_mode_gain self.couplings['out_mode_gain'] = out_mode_gain # Reduce uvlm projected onto structural coordinates if uvlm.rom: if rigid_dof != 0: self.runrom_rbm(uvlm) else: for k, rom in uvlm.rom.items(): uvlm.ss = rom.run(uvlm.ss) else: uvlm.ss = self.load_uvlm(self.settings['uvlm_filename']) # Coupling matrices Tas = libss.Gain(np.eye(uvlm.ss.inputs, beam.ss.outputs), input_vars=LinearVector.transform(beam.ss.output_variables, to_type=InputVariable), output_vars=LinearVector.transform(uvlm.ss.input_variables, to_type=OutputVariable)) Tsa = libss.Gain(np.eye(beam.ss.inputs, uvlm.ss.outputs), input_vars=LinearVector.transform(uvlm.ss.output_variables, to_type=InputVariable), output_vars=LinearVector.transform(beam.ss.input_variables, to_type=OutputVariable)) # Scale coupling matrices if uvlm.scaled: Tsa.value *= uvlm.sys.ScalingFacts['force'] * uvlm.sys.ScalingFacts['time'] ** 2 if rigid_dof > 0: Tas.value[:flex_nodes + 6, :flex_nodes + 6] /= uvlm.sys.ScalingFacts['length'] Tas.value[total_dof: total_dof + flex_nodes + 6] /= uvlm.sys.ScalingFacts['length'] else: if not self.settings['beam_settings']['modal_projection']: Tas.value /= uvlm.sys.ScalingFacts['length'] ss = libss.couple(ss01=uvlm.ss, ss02=beam.ss, K12=Tas, K21=Tsa) self.couplings['Tas'] = Tas self.couplings['Tsa'] = Tsa self.state_variables = {'aero': uvlm.ss.states, 'beam': beam.ss.states} # Save zero force reference self.linearisation_vectors['forces_aero_beam_dof'] = Ksa.dot(self.linearisation_vectors['forces_aero']) if self.settings['beam_settings']['modal_projection']: self.linearisation_vectors['mode_shapes'] = beam.sys.U if self.settings['beam_settings']['modal_projection'] is True and \ self.settings['beam_settings']['inout_coords'] == 'modes': self.linearisation_vectors['forces_aero_beam_dof'] = out_mode_matrix.dot(self.linearisation_vectors['forces_aero_beam_dof']) cout.cout_wrap('Aeroelastic system assembled:') cout.cout_wrap('\tAerodynamic states: %g' % uvlm.ss.states, 1) cout.cout_wrap('\tStructural states: %g' % beam.ss.states, 1) cout.cout_wrap('\tTotal states: %g' % ss.states, 1) cout.cout_wrap('\tInputs: %g' % ss.inputs, 1) cout.cout_wrap('\tOutputs: %g' % ss.outputs, 1) self.ss = ss return self.ss
[docs] def update(self, u_infty): """ Updates the aeroelastic scaled system with the new reference velocity. Only the beam equations need updating since the only dependency in the forward flight velocity resides there. Args: u_infty (float): New reference velocity Returns: sharpy.linear.src.libss.StateSpace: Updated aeroelastic state-space system """ t_ref = self.uvlm.sys.ScalingFacts['length'] / u_infty self.beam.sys.update_matrices_time_scale(t_ref) self.beam.sys.assemble() if self.beam.sys.SSdisc is not None: self.beam.ss = self.beam.sys.SSdisc elif self.beam.sys.SScont is not None: self.beam.ss = self.beam.sys.SScont else: raise AttributeError('Could not find either a continuous or discrete system in Beam') self.ss = libss.couple(ss01=self.uvlm.ss, ss02=self.beam.ss, K12=self.couplings['Tas'], K21=self.couplings['Tsa']) return self.ss
def runrom_rbm(self, uvlm): ss = uvlm.ss rig_nodes = self.beam.sys.num_dof_rig if rig_nodes == 9: orient_dof = 3 else: orient_dof = 4 # Input side if self.settings['beam_settings']['modal_projection'] is True and \ self.settings['beam_settings']['inout_coords'] == 'modes': rem_int_modes = np.zeros((ss.inputs, ss.inputs - rig_nodes)) rem_int_modes[rig_nodes:, :] = np.eye(ss.inputs - rig_nodes) # Output side - remove quaternion equations output rem_quat_out = np.zeros((ss.outputs-orient_dof, ss.outputs)) # find quaternion indices U = self.beam.sys.U indices = np.where(U[-orient_dof:, :] == 1.)[1] j = 0 for i in range(ss.outputs): if i in indices: continue rem_quat_out[j, i] = 1 j += 1 in_vars = ss.input_variables.copy() in_vars.modify('q', size=in_vars.get_variable_from_name('q').size - rig_nodes) remove_integro_inputs = libss.Gain(rem_int_modes, input_vars=in_vars, output_vars=libss.LinearVector.transform(ss.input_variables, to_type=OutputVariable)) out_vars = ss.output_variables.copy() out_vars.modify('Q', size=out_vars.get_variable_from_name('Q').size-orient_dof) remove_quaternion_out = libss.Gain(rem_quat_out, input_vars=libss.LinearVector.transform(ss.output_variables, to_type=InputVariable), output_vars=out_vars) else: # TODO: THESE NEED DOING rem_int_modes = np.zeros((ss.inputs, ss.inputs - rig_nodes)) rem_int_modes[:self.beam.sys.num_dof_flex, :self.beam.sys.num_dof_flex] = \ np.eye(self.beam.sys.num_dof_flex) rem_int_modes[self.beam.sys.num_dof_flex+rig_nodes:, self.beam.sys.num_dof_flex:] = \ np.eye(ss.inputs - self.beam.sys.num_dof_flex - rig_nodes) rem_quat_out = np.zeros((ss.outputs-orient_dof, ss.outputs)) rem_quat_out[:, :-orient_dof] = np.eye(ss.outputs-orient_dof) ss.addGain(remove_integro_inputs, where='in') ss.addGain(remove_quaternion_out, where='out') for k, rom in uvlm.rom.items(): uvlm.ss = rom.run(uvlm.ss) add_integro_inputs = remove_integro_inputs.transpose() add_quaternion_outputs = remove_quaternion_out.transpose() uvlm.ss.addGain(add_integro_inputs, where='in') uvlm.ss.addGain(add_quaternion_outputs, where='out')
[docs] def get_gebm2uvlm_gains(self, data): r""" Provides: - the gain matrices required to connect the linearised GEBM and UVLM inputs/outputs - the stiffening and damping factors to be added to the linearised GEBM equations in order to account for non-zero aerodynamic loads at the linearisation point. The function produces the gain matrices: - ``Kdisp``: gains from GEBM to UVLM grid displacements - ``Kvel_disp``: influence of GEBM dofs displacements to UVLM grid velocities. - ``Kvel_vel``: influence of GEBM dofs displacements to UVLM grid displacements. - ``Kforces`` (UVLM->GEBM) dimensions are the transpose than the Kdisp and Kvel* matrices. Hence, when allocation this term, ``ii`` and ``jj`` indices will unintuitively refer to columns and rows, respectively. And the stiffening/damping terms accounting for non-zero aerodynamic forces at the linearisation point: - ``Kss``: stiffness factor (flexible dof -> flexible dof) accounting for non-zero forces at the linearisation point. - ``Csr``: damping factor (rigid dof -> flexible dof) - ``Crs``: damping factor (flexible dof -> rigid dof) - ``Crr``: damping factor (rigid dof -> rigid dof) Stiffening and damping related terms due to the non-zero aerodynamic forces at the linearisation point: .. math:: \mathbf{F}_{A,n} = C^{AG}(\mathbf{\chi})\sum_j \mathbf{f}_{G,j} \rightarrow \delta\mathbf{F}_{A,n} = C^{AG}_0 \sum_j \delta\mathbf{f}_{G,j} + \frac{\partial}{\partial\chi}(C^{AG}\sum_j \mathbf{f}_{G,j}^0)\delta\chi The term multiplied by the variation in the quaternion, :math:`\delta\chi`, couples the forces with the rigid body equations and becomes part of :math:`\mathbf{C}_{sr}`. Similarly, the linearisation of the moments results in expression that contribute to the stiffness and damping matrices. .. math:: \mathbf{M}_{B,n} = \sum_j \tilde{X}_B C^{BA}(\Psi)C^{AG}(\chi)\mathbf{f}_{G,j} .. math:: \delta\mathbf{M}_{B,n} = \sum_j \tilde{X}_B\left(C_0^{BG}\delta\mathbf{f}_{G,j} + \frac{\partial}{\partial\Psi}(C^{BA}\delta\mathbf{f}^0_{A,j})\delta\Psi + \frac{\partial}{\partial\chi}(C^{BA}_0 C^{AG} \mathbf{f}_{G,j})\delta\chi\right) The linearised equations of motion for the geometrically exact beam model take the input term :math:`\delta \mathbf{Q}_n = \{\delta\mathbf{F}_{A,n},\, T_0^T\delta\mathbf{M}_{B,n}\}`, which means that the moments should be provided as :math:`T^T(\Psi)\mathbf{M}_B` instead of :math:`\mathbf{M}_A = C^{AB}\mathbf{M}_B`, where :math:`T(\Psi)` is the tangential operator. .. math:: \delta(T^T\mathbf{M}_B) = T^T_0\delta\mathbf{M}_B + \frac{\partial}{\partial\Psi}(T^T\delta\mathbf{M}_B^0)\delta\Psi is the linearised expression for the moments, where the first term would correspond to the input terms to the beam equations and the second arises due to the non-zero aerodynamic moment at the linearisation point and must be subtracted (since it comes from the forces) to form part of :math:`\mathbf{K}_{ss}`. In addition, the :math:`\delta\mathbf{M}_B` term depends on both :math:`\delta\Psi` and :math:`\delta\chi`, therefore those terms would also contribute to :math:`\mathbf{K}_{ss}` and :math:`\mathbf{C}_{sr}`, respectively. The contribution from the total forces and moments will be accounted for in :math:`\mathbf{C}_{rr}` and :math:`\mathbf{C}_{rs}`. .. math:: \delta\mathbf{F}_{tot,A} = \sum_n\left(C^{GA}_0 \sum_j \delta\mathbf{f}_{G,j} + \frac{\partial}{\partial\chi}(C^{AG}\sum_j \mathbf{f}_{G,j}^0)\delta\chi\right) Therefore, after running this method, the beam matrices will be updated as: >>> K_beam[:flex_dof, :flex_dof] += Kss >>> C_beam[:flex_dof, -rigid_dof:] += Csr >>> C_beam[-rigid_dof:, :flex_dof] += Crs >>> C_beam[-rigid_dof:, -rigid_dof:] += Crr Track body option The ``track_body`` setting restricts the UVLM grid to linear translation motions and therefore should be used to ensure that the forces are computed using the reference linearisation frame. The UVLM and beam are linearised about a reference equilibrium condition. The UVLM is defined in the inertial reference frame while the beam employs the body attached frame and therefore a projection from one frame onto another is required during the coupling process. However, the inputs to the UVLM (i.e. the lattice grid coordinates) are obtained from the beam deformation which is expressed in A frame and therefore the grid coordinates need to be projected onto the inertial frame ``G``. As the beam rotates, the projection onto the ``G`` frame of the lattice grid coordinates will result in a grid that is not coincident with that at the linearisation reference and therefore the grid coordinates must be projected onto the original frame, which will be referred to as ``U``. The transformation between the inertial frame ``G`` and the ``U`` frame is a function of the rotation of the ``A`` frame and the original position: .. math:: C^{UG}(\chi) = C^{GA}(\chi_0)C^{AG}(\chi) Therefore, the grid coordinates obtained in ``A`` frame and projected onto the ``G`` frame can be transformed to the ``U`` frame using .. math:: \zeta_U = C^{UG}(\chi) \zeta_G which allows the grid lattice coordinates to be projected onto the original linearisation frame. In a similar fashion, the output lattice vertex forces of the UVLM are defined in the original linearisation frame ``U`` and need to be transformed onto the inertial frame ``G`` prior to projecting them onto the ``A`` frame to use them as the input forces to the beam system. .. math:: \boldsymbol{f}_G = C^{GU}(\chi)\boldsymbol{f}_U The linearisation of the above relations lead to the following expressions that have to be added to the coupling matrices: * ``Kdisp_vel`` terms: .. math:: \delta\boldsymbol{\zeta}_U= C^{GA}_0 \frac{\partial}{\partial \boldsymbol{\chi}} \left(C^{AG}\boldsymbol{\zeta}_{G,0}\right)\delta\boldsymbol{\chi} + \delta\boldsymbol{\zeta}_G * ``Kvel_vel`` terms: .. math:: \delta\dot{\boldsymbol{\zeta}}_U= C^{GA}_0 \frac{\partial}{\partial \boldsymbol{\chi}} \left(C^{AG}\dot{\boldsymbol{\zeta}}_{G,0}\right)\delta\boldsymbol{\chi} + \delta\dot{\boldsymbol{\zeta}}_G The transformation of the forces and moments introduces terms that are functions of the orientation and are included as stiffening and damping terms in the beam's matrices: * ``Csr`` damping terms relating to translation forces: .. math:: C_{sr}^{tra} -= \frac{\partial}{\partial\boldsymbol{\chi}} \left(C^{GA} C^{AG}_0 \boldsymbol{f}_{G,0}\right)\delta\boldsymbol{\chi} * ``Csr`` damping terms related to moments: .. math:: C_{sr}^{rot} -= T^\top\widetilde{\mathbf{X}}_B C^{BG} \frac{\partial}{\partial\boldsymbol{\chi}} \left(C^{GA} C^{AG}_0 \boldsymbol{f}_{G,0}\right)\delta\boldsymbol{\chi} The ``track_body`` setting. When ``track_body`` is enabled, the UVLM grid is no longer coincident with the inertial reference frame throughout the simulation but rather it is able to rotate as the ``A`` frame rotates. This is to simulate a free flying vehicle, where, for instance, the orientation does not affect the aerodynamics. The UVLM defined in this frame of reference, named ``U``, satisfies the following convention: * The ``U`` frame is coincident with the ``G`` frame at the time of linearisation. * The ``U`` frame rotates as the ``A`` frame rotates. Transformations related to the ``U`` frame of reference: * The angle between the ``U`` frame and the ``A`` frame is always constant and equal to :math:`\boldsymbol{\Theta}_0`. * The angle between the ``A`` frame and the ``G`` frame is :math:`\boldsymbol{\Theta}=\boldsymbol{\Theta}_0 + \delta\boldsymbol{\Theta}` * The projection of a vector expressed in the ``G`` frame onto the ``U`` frame is expressed by: .. math:: \boldsymbol{v}^U = C^{GA}_0 C^{AG} \boldsymbol{v}^G * The reverse, a projection of a vector expressed in the ``U`` frame onto the ``G`` frame, is expressed by .. math:: \boldsymbol{v}^U = C^{GA} C^{AG}_0 \boldsymbol{v}^U The effect this has on the aeroelastic coupling between the UVLM and the structural dynamics is that the orientation and change of orientation of the vehicle has no effect on the aerodynamics. The aerodynamics are solely affected by the contribution of the 6-rigid body velocities (as well as the flexible DOFs velocities). """ aero = data.aero structure = data.structure tsaero = self.uvlm.tsaero0 tsstr = self.beam.tsstruct0 Kzeta = self.uvlm.sys.Kzeta num_dof_str = self.beam.sys.num_dof_str num_dof_rig = self.beam.sys.num_dof_rig num_dof_flex = self.beam.sys.num_dof_flex use_euler = self.beam.sys.use_euler # allocate output Kdisp = np.zeros((3 * Kzeta, num_dof_str)) Kdisp_vel = np.zeros((3 * Kzeta, num_dof_str)) # Orientation is in velocity DOFs Kvel_disp = np.zeros((3 * Kzeta, num_dof_str)) Kvel_vel = np.zeros((3 * Kzeta, num_dof_str)) Kforces = np.zeros((num_dof_str, 3 * Kzeta)) Kss = np.zeros((num_dof_flex, num_dof_flex)) Csr = np.zeros((num_dof_flex, num_dof_rig)) Crs = np.zeros((num_dof_rig, num_dof_flex)) Crr = np.zeros((num_dof_rig, num_dof_rig)) Krs = np.zeros((num_dof_rig, num_dof_flex)) # get projection matrix A->G # (and other quantities indep. from nodal position) Cga = algebra.quat2rotation(tsstr.quat) # NG 6-8-19 removing .T Cag = Cga.T # for_pos=tsstr.for_pos for_vel = tsstr.for_vel[:3] for_rot = tsstr.for_vel[3:] skew_for_rot = algebra.skew(for_rot) Der_vel_Ra = np.dot(Cga, skew_for_rot) Faero = np.zeros(3) FaeroA = np.zeros(3) # GEBM degrees of freedom jj_for_tra = range(num_dof_str - num_dof_rig, num_dof_str - num_dof_rig + 3) jj_for_rot = range(num_dof_str - num_dof_rig + 3, num_dof_str - num_dof_rig + 6) if use_euler: jj_euler = range(num_dof_str - 3, num_dof_str) euler = algebra.quat2euler(tsstr.quat) tsstr.euler = euler else: jj_quat = range(num_dof_str - 4, num_dof_str) jj = 0 # nodal dof index for node_glob in range(structure.num_node): ### detect bc at node (and no. of dofs) bc_here = structure.boundary_conditions[node_glob] if bc_here == 1: # clamp (only rigid-body) dofs_here = 0 jj_tra, jj_rot = [], [] # continue elif bc_here == -1 or bc_here == 0: # (rigid+flex body) dofs_here = 6 jj_tra = 6 * structure.vdof[node_glob] + np.array([0, 1, 2], dtype=int) jj_rot = 6 * structure.vdof[node_glob] + np.array([3, 4, 5], dtype=int) else: raise NameError('Invalid boundary condition (%d) at node %d!' \ % (bc_here, node_glob)) jj += dofs_here # retrieve element and local index ee, node_loc = structure.node_master_elem[node_glob, :] # get position, crv and rotation matrix Ra = tsstr.pos[node_glob, :] # in A FoR, w.r.t. origin A-G Rg = np.dot(Cag.T, Ra) # in G FoR, w.r.t. origin A-G psi = tsstr.psi[ee, node_loc, :] psi_dot = tsstr.psi_dot[ee, node_loc, :] Cab = algebra.crv2rotation(psi) Cba = Cab.T Cbg = np.dot(Cab.T, Cag) Tan = algebra.crv2tan(psi) track_body = self.settings['track_body'] ### str -> aero mapping # some nodes may be linked to multiple surfaces... for str2aero_here in aero.struct2aero_mapping[node_glob]: # detect surface/span-wise coordinate (ss,nn) nn, ss = str2aero_here['i_n'], str2aero_here['i_surf'] # print('%.2d,%.2d'%(nn,ss)) # surface panelling M = aero.aero_dimensions[ss][0] N = aero.aero_dimensions[ss][1] Kzeta_start = 3 * sum(self.uvlm.sys.MS.KKzeta[:ss]) shape_zeta = (3, M + 1, N + 1) for mm in range(M + 1): # get bound vertex index ii_vert = [Kzeta_start + np.ravel_multi_index( (cc, mm, nn), shape_zeta) for cc in range(3)] # get position vectors zetag = tsaero.zeta[ss][:, mm, nn] # in G FoR, w.r.t. origin A-G zetaa = np.dot(Cag, zetag) # in A FoR, w.r.t. origin A-G Xg = zetag - Rg # in G FoR, w.r.t. origin B Xb = np.dot(Cbg, Xg) # in B FoR, w.r.t. origin B # get rotation terms Xbskew = algebra.skew(Xb) XbskewTan = np.dot(Xbskew, Tan) # get velocity terms zetag_dot = tsaero.zeta_dot[ss][:, mm, nn] - Cga.dot(for_vel) # in G FoR, w.r.t. origin A-G zetaa_dot = np.dot(Cag, zetag_dot) # in A FoR, w.r.t. origin A-G # get aero force faero = tsaero.forces[ss][:3, mm, nn] Faero += faero faero_a = np.dot(Cag, faero) FaeroA += faero_a maero_g = np.cross(Xg, faero) maero_b = np.dot(Cbg, maero_g) ### ---------------------------------------- allocate Kdisp if bc_here != 1: # wrt pos - Eq 25 second term Kdisp[np.ix_(ii_vert, jj_tra)] += Cga # wrt psi - Eq 26 Kdisp[np.ix_(ii_vert, jj_rot)] -= np.dot(Cbg.T, XbskewTan) # w.r.t. position of FoR A (w.r.t. origin G) # null as A and G have always same origin in SHARPy # # ### w.r.t. quaternion (attitude changes) if use_euler: Kdisp_vel[np.ix_(ii_vert, jj_euler)] += \ algebra.der_Ceuler_by_v(tsstr.euler, zetaa) # Track body - project inputs as for A not moving if track_body: Kdisp_vel[np.ix_(ii_vert, jj_euler)] += \ Cga.dot(algebra.der_Peuler_by_v(tsstr.euler, zetag)) else: # Equation 25 # Kdisp[np.ix_(ii_vert, jj_quat)] += \ # algebra.der_Cquat_by_v(tsstr.quat, zetaa) Kdisp_vel[np.ix_(ii_vert, jj_quat)] += \ algebra.der_Cquat_by_v(tsstr.quat, zetaa) # Track body - project inputs as for A not moving if track_body: Kdisp_vel[np.ix_(ii_vert, jj_quat)] += \ Cga.dot(algebra.der_CquatT_by_v(tsstr.quat, zetag)) ### ------------------------------------ allocate Kvel_disp if bc_here != 1: # # wrt pos Kvel_disp[np.ix_(ii_vert, jj_tra)] += Der_vel_Ra # wrt psi (at zero psi_dot) Kvel_disp[np.ix_(ii_vert, jj_rot)] -= \ np.dot(Cga, np.dot(skew_for_rot, np.dot(Cab, XbskewTan))) # # wrt psi (psi_dot contributions - verified) Kvel_disp[np.ix_(ii_vert, jj_rot)] += np.dot(Cbg.T, np.dot( algebra.skew(np.dot(XbskewTan, psi_dot)), Tan)) if np.linalg.norm(psi) >= 1e-6: Kvel_disp[np.ix_(ii_vert, jj_rot)] -= \ np.dot(Cbg.T, np.dot(Xbskew, algebra.der_Tan_by_xv(psi, psi_dot))) # # w.r.t. position of FoR A (w.r.t. origin G) # # null as A and G have always same origin in SHARPy # # ### w.r.t. quaternion (attitude changes) - Eq 30 if use_euler: Kvel_vel[np.ix_(ii_vert, jj_euler)] += \ algebra.der_Ceuler_by_v(tsstr.euler, zetaa_dot) # Track body if ForA is rotating if track_body: Kvel_vel[np.ix_(ii_vert, jj_euler)] += \ Cga.dot(algebra.der_Peuler_by_v(tsstr.euler, zetag_dot)) else: Kvel_vel[np.ix_(ii_vert, jj_quat)] += \ algebra.der_Cquat_by_v(tsstr.quat, zetaa_dot) # Track body if ForA is rotating if track_body: Kvel_vel[np.ix_(ii_vert, jj_quat)] += \ Cga.dot(algebra.der_CquatT_by_v(tsstr.quat, zetag_dot)) ### ------------------------------------- allocate Kvel_vel if bc_here != 1: # wrt pos_dot Kvel_vel[np.ix_(ii_vert, jj_tra)] += Cga # # wrt crv_dot Kvel_vel[np.ix_(ii_vert, jj_rot)] -= np.dot(Cbg.T, XbskewTan) # # wrt velocity of FoR A Kvel_vel[np.ix_(ii_vert, jj_for_tra)] += Cga Kvel_vel[np.ix_(ii_vert, jj_for_rot)] -= \ np.dot(Cga, algebra.skew(zetaa)) # wrt rate of change of quaternion: not implemented! ### -------------------------------------- allocate Kforces if bc_here != 1: # nodal forces Kforces[np.ix_(jj_tra, ii_vert)] += Cag # nodal moments Kforces[np.ix_(jj_rot, ii_vert)] += \ np.dot(Tan.T, np.dot(Cbg, algebra.skew(Xg))) # or, equivalently, np.dot( algebra.skew(Xb),Cbg) # total forces Kforces[np.ix_(jj_for_tra, ii_vert)] += Cag # total moments Kforces[np.ix_(jj_for_rot, ii_vert)] += \ np.dot(Cag, algebra.skew(zetag)) # quaternion equation # null, as not dep. on external forces ### --------------------------------------- allocate Kstiff ### flexible dof equations (Kss and Csr) if bc_here != 1: # nodal forces if use_euler: if not track_body: Csr[jj_tra, -3:] -= algebra.der_Peuler_by_v(tsstr.euler, faero) # Csr[jj_tra, -3:] -= algebra.der_Ceuler_by_v(tsstr.euler, Cga.T.dot(faero)) else: if not track_body: Csr[jj_tra, -4:] -= algebra.der_CquatT_by_v(tsstr.quat, faero) # Track body # if track_body: # Csr[jj_tra, -4:] -= algebra.der_Cquat_by_v(tsstr.quat, Cga.T.dot(faero)) ### moments TanTXbskew = np.dot(Tan.T, Xbskew) # contrib. of TanT (dpsi) - Eq 37 - Integration of UVLM and GEBM Kss[np.ix_(jj_rot, jj_rot)] -= algebra.der_TanT_by_xv(psi, maero_b) # contrib of delta aero moment (dpsi) - Eq 36 Kss[np.ix_(jj_rot, jj_rot)] -= \ np.dot(TanTXbskew, algebra.der_CcrvT_by_v(psi, np.dot(Cag, faero))) # contribution of delta aero moment (dquat) if use_euler: if not track_body: Csr[jj_rot, -3:] -= \ np.dot(TanTXbskew, np.dot(Cba, algebra.der_Peuler_by_v(tsstr.euler, faero))) # if track_body: # Csr[jj_rot, -3:] -= \ # np.dot(TanTXbskew, # np.dot(Cbg, # algebra.der_Peuler_by_v(tsstr.euler, Cga.T.dot(faero)))) else: if not track_body: Csr[jj_rot, -4:] -= \ np.dot(TanTXbskew, np.dot(Cba, algebra.der_CquatT_by_v(tsstr.quat, faero))) # Track body # if track_body: # Csr[jj_rot, -4:] -= \ # np.dot(TanTXbskew, # np.dot(Cbg, # algebra.der_CquatT_by_v(tsstr.quat, Cga.T.dot(faero)))) ### rigid body eqs (Crs and Crr) if bc_here != 1: # Changed Crs to Krs - NG 14/5/19 # moments contribution due to delta_Ra (+ sign intentional) Krs[3:6, jj_tra] += algebra.skew(faero_a) # moment contribution due to delta_psi (+ sign intentional) Krs[3:6, jj_rot] += np.dot(algebra.skew(faero_a), algebra.der_Ccrv_by_v(psi, Xb)) if use_euler: if not track_body: # total force Crr[:3, -3:] -= algebra.der_Peuler_by_v(tsstr.euler, faero) # total moment contribution due to change in euler angles Crr[3:6, -3:] -= algebra.der_Peuler_by_v(tsstr.euler, np.cross(zetag, faero)) Crr[3:6, -3:] += np.dot( np.dot(Cag, algebra.skew(faero)), algebra.der_Peuler_by_v(tsstr.euler, np.dot(Cab, Xb))) else: if not track_body: # total force Crr[:3, -4:] -= algebra.der_CquatT_by_v(tsstr.quat, faero) # total moment contribution due to quaternion Crr[3:6, -4:] -= algebra.der_CquatT_by_v(tsstr.quat, np.cross(zetag, faero)) Crr[3:6, -4:] += np.dot( np.dot(Cag, algebra.skew(faero)), algebra.der_CquatT_by_v(tsstr.quat, np.dot(Cab, Xb))) # # Track body # if track_body: # # NG 20/8/19 - is the Cag needed here? Verify # Crr[:3, -4:] -= Cag.dot(algebra.der_Cquat_by_v(tsstr.quat, Cga.T.dot(faero))) # # Crr[3:6, -4:] -= Cag.dot(algebra.skew(zetag).dot(algebra.der_Cquat_by_v(tsstr.quat, Cga.T.dot(faero)))) # Crr[3:6, -4:] += Cag.dot(algebra.skew(faero)).dot(algebra.der_Cquat_by_v(tsstr.quat, Cga.T.dot(zetag))) # transfer self.Kdisp = Kdisp self.Kvel_disp = Kvel_disp self.Kdisp_vel = Kdisp_vel self.Kvel_vel = Kvel_vel self.Kforces = Kforces # stiffening factors self.Kss = Kss self.Krs = Krs self.Csr = Csr self.Crs = Crs self.Crr = Crr
[docs] def to_nodal_coordinates(self): """ Transforms the outputs of the system to nodal coordinates if they were previously expressed in modal space """ is_modal = self.beam.sys.modal if is_modal: beam_kin = self.beam.sys.Kin # N to Q beam_kout = self.beam.sys.Kout # q to eta aug_in_gain = sclalg.block_diag(self.couplings['in_mode_gain'].value.T, beam_kin.value) input_gain = libss.Gain(aug_in_gain, input_vars=LinearVector.merge( LinearVector.transform( self.couplings['in_mode_gain'].output_variables, to_type=InputVariable), beam_kin.input_variables), output_vars=LinearVector.merge( LinearVector.transform( self.couplings['in_mode_gain'].input_variables, to_type=OutputVariable), beam_kin.output_variables) ) try: acceleration_gain = self.beam.sys.acceleration_modal_gain except AttributeError: aug_out_gain = sclalg.block_diag(self.couplings['out_mode_gain'].value.T, beam_kout.value) output_gain = libss.Gain(aug_out_gain, input_vars=LinearVector.merge( LinearVector.transform( self.couplings['out_mode_gain'].output_variables, to_type=InputVariable), beam_kout.input_variables), output_vars=LinearVector.merge( LinearVector.transform( self.couplings['out_mode_gain'].input_variables, to_type=OutputVariable), beam_kout.output_variables) ) else: aug_out_gain = sclalg.block_diag(self.couplings['out_mode_gain'].value.T, beam_kout.value, acceleration_gain.value) output_gain = libss.Gain(aug_out_gain, input_vars=LinearVector.merge( LinearVector.transform( self.couplings['out_mode_gain'].output_variables, to_type=InputVariable), LinearVector.merge(beam_kout.input_variables, acceleration_gain.input_variables), ), output_vars=LinearVector.merge( LinearVector.transform( self.couplings['out_mode_gain'].input_variables, to_type=OutputVariable), LinearVector.merge(beam_kout.output_variables, acceleration_gain.output_variables)) ) self.ss.addGain(input_gain, where='in') self.ss.addGain(output_gain, where='out')
@staticmethod def load_uvlm(filename): import sharpy.utils.h5utils as h5 cout.cout_wrap('Loading UVLM state space system projected onto structural DOFs from file') read_data = h5.readh5(filename).ss # uvlm_ss_read = read_data.linear.linear_system.uvlm.ss uvlm_ss_read = read_data return libss.StateSpace(uvlm_ss_read.A, uvlm_ss_read.B, uvlm_ss_read.C, uvlm_ss_read.D, dt=uvlm_ss_read.dt)