sdynpy.signal_processing.sdynpy_massprops.rigid_body_shape_scaling

• Module: sdynpy.signal_processing.sdynpy_massprops

• Source: GitHub

Signature¶

def sdynpy.signal_processing.sdynpy_massprops.rigid_body_shape_scaling(rbm_modes, trans_shapes, rot_shapes, center_of_rotation=None, return_mass_properties=False, rcond=None, symmetrize=True, check_rank=True)

Recover scaled/mass-normalized rigid-body shapes and optionally mass properties from a set of mass-normalized rigid-body modes and user-supplied pure rigid-body translation/rotation shapes.

Parameters¶

• rbm_modes : (n, 6) array_like Mass-normalized rigid-body mode shapes. Columns span the 6 rigid-body modes. Assumed to satisfy Phi.T @ M @ Phi = I in the underlying mass metric.

• trans_shapes : (n, 3) array_like Unscaled pure translation shapes corresponding to unit translations in x, y, z (or equivalent unit physical translations in the chosen measurement basis).

• rot_shapes : (n, 3) array_like Unscaled pure rotation shapes corresponding to unit rotations about x, y, z (1 radian), defined about some reference point.

• center_of_rotation : (3,) array_like, optional Reference point about which rot_shapes were constructed. If provided, the center of mass is also returned in global coordinates. If omitted, COM is returned only relative to the rotation reference point.

• return_mass_properties : bool, optional If True, also return recovered mass properties.

• rcond : float, optional Cutoff for pseudoinverse.

• symmetrize : bool, optional If True, symmetrize the recovered rigid-body mass matrix and inertia tensors.

• check_rank : bool, optional If True, check that the supplied rigid-body basis has rank 6.

Returns¶

• scaled_shapes : (n, 6) ndarray Mass-normalized/scaled physical rigid-body shapes corresponding as closely as possible to the supplied pure translation/rotation basis. Columns correspond to [Tx, Ty, Tz, Rx, Ry, Rz].

• mass_props : dict, optional Returned only if return_mass_properties=True. Contains:

  • ‘A’ : (6, 6) transformation matrix such that B ≈ Phi @ A

  • ‘G’ : (6, 6) rigid-body generalized mass matrix in basis B

  • ‘mass’ : scalar total mass

  • ‘com_relative’ : (3,) center of mass relative to rotation reference point

  • ‘com_global’ : (3,) center of mass in global coordinates (if center_of_rotation provided)

  • ‘inertia_about_reference’ : (3, 3) inertia tensor about the rotation reference point

  • ‘inertia_about_com’ : (3, 3) inertia tensor about center of mass

  • ‘principal_inertias’ : (3,) eigenvalues of inertia_about_com

  • ‘principal_axes’ : (3, 3) eigenvectors of inertia_about_com

  • ‘reconstruction_error’ : relative Frobenius norm of B - Phi @ A

  • ‘G_tt’, ‘G_tr’, ‘G_rt’, ‘G_rr’ : 3x3 blocks of G

Notes¶

Let Phi = rbm_modes and B = [trans_shapes rot_shapes]. Since both span the rigid-body subspace, B = Phi A for some A. Using mass normalization Phi.T M Phi = I, the rigid-body generalized mass matrix is

G = B.T M B = A.T A

The mass properties are extracted from:

G = [[ m I,      -m [c]_x ],
     [ m [c]_x,   J_ref   ]]

where c is COM relative to the reference point used to define the rotation shapes, and J_ref is the inertia tensor about that same reference point.

Sign convention for c assumes the rotation shapes follow: u = omega x (x - O)