Skip to content

visualdynamics.demo.plate

plate

The demonstration test article: a free aluminum plate.

A 12 in x 12 in x 0.5 in plate of 6061-T6, hung free-free — the canonical modal-survey article, chosen because a completely free square plate is the classical benchmark: its nondimensional frequencies are tabulated (Leissa, NASA SP-160) and every test engineer has rapped one with a hammer. Where the drone demonstrates scale, the plate demonstrates truth: its answers can be checked against a book.

Built entirely with visualdynamics.fem — the rectangular MITC4 shell element, validated against the same tables in tests/test_fem.py — so it owes nothing to any other library. It is the model the small test fixtures come from, the way the drone is the model the large ones come from.

from visualdynamics.demo import plate

model = plate.build()
shapes = model.eigensolution(maximum_frequency=2500, damping=0.02)

Run it directly to print what it weighs and where its modes land against the classical table:

python3 -m visualdynamics.demo.plate

The article is defined in inches and pounds — the drawing a machine shop would get — and carried in SI exactly, converted once below, because fem.Model is SI inside like everything else in the package. Display units are the status bar's business.

The plate answers to the tables to about a percent, not exactly, for two honest reasons: the mesh (discretization converges from above as h^2 — tests/test_fem.py measures it per mode), and Poisson's ratio (the classical values are computed at 0.3, 6061 is 0.33, and a free plate's frequencies shift a little with it).

Functions:

Name Description
build

The plate, meshed mesh x mesh.
instrumented

The nodes a modal survey of this plate would put sensors on.
describe

What it weighs and where its modes land against the book.

Functions:

build 

build(mesh: int = MESH) -> Model

The plate, meshed mesh x mesh.

Node ids read like a survey grid: row 1 is 101..1xx along the front edge, row 2 is 201.., so node 704 is four points along the seventh row — the numbering a channel table would use.

Source code in src/visualdynamics/demo/plate.py 
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
def build(mesh: int = MESH) -> fem.Model:
    """The plate, meshed `mesh` x `mesh`.

    Node ids read like a survey grid: row 1 is 101..1xx along the
    front edge, row 2 is 201.., so node 704 is four points along the
    seventh row — the numbering a channel table would use.
    """
    model = fem.Model('plate', length_unit='m')
    for j in range(mesh + 1):
        for i in range(mesh + 1):
            model.add_node((j + 1) * 100 + (i + 1),
                           i * SIDE / mesh, j * SIDE / mesh, 0.0)
    for j in range(mesh):
        for i in range(mesh):
            corner = (j + 1) * 100 + (i + 1)
            model.add_plate((corner, corner + 1,
                             corner + 101, corner + 100),
                            ALUMINUM, THICKNESS)
    return model

instrumented 

instrumented(model: Model) -> dict[str, list[int]]

The nodes a modal survey of this plate would put sensors on.

A five-by-five grid of accelerometers — the layout a real plate survey uses, dense enough that the sixth elastic mode (three half-waves) is unaliased — and the drive point at a corner, because a corner of a free plate moves in every low mode while the centre sits on the node lines of all the antisymmetric ones. Found from the model rather than written down, for the same reason the drone's are: numbering follows the mesh.

Source code in src/visualdynamics/demo/plate.py 
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
def instrumented(model: fem.Model) -> dict[str, list[int]]:
    """The nodes a modal survey of this plate would put sensors on.

    A five-by-five grid of accelerometers — the layout a real plate
    survey uses, dense enough that the sixth elastic mode (three
    half-waves) is unaliased — and the drive point at a corner,
    because a corner of a free plate moves in every low mode while
    the centre sits on the node lines of all the antisymmetric ones.
    Found from the model rather than written down, for the same
    reason the drone's are: numbering follows the mesh.
    """
    ids = model.node_ids
    rows = round(math.sqrt(len(ids))) - 1
    step = rows // 4 if rows >= 4 else 1
    stations = sorted({0, step, 2 * step, 3 * step, rows})
    grid = [(j + 1) * 100 + (i + 1)
            for j in stations for i in stations]
    return {'grid': grid, 'drive': [grid[0]]}

describe 

describe(model: Model | None = None) -> None

What it weighs and where its modes land against the book.

Source code in src/visualdynamics/demo/plate.py 
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
def describe(model: fem.Model | None = None) -> None:
    """What it weighs and where its modes land against the book."""
    model = model or build()
    shapes = model.eigensolution(maximum_frequency=2500.0)
    bending = (ALUMINUM.youngs_modulus * THICKNESS ** 3
               / (12.0 * (1.0 - ALUMINUM.poissons_ratio ** 2)))
    scale = (2.0 * math.pi * SIDE ** 2
             * math.sqrt(ALUMINUM.density * THICKNESS / bending))
    classical = [13.468, 19.596, 24.271, 34.801, 34.801, 61.093]

    pounds = model.structural_mass / 0.45359237
    print(f'{model.num_nodes} nodes, {model.num_dof} DOF, '
          f'{len(model.plates)} plate elements')
    print(f'{model.structural_mass:.2f} kg ({pounds:.1f} lb)')
    print()
    print(f'{"f (Hz)":>9}  {"lambda^2":>9}  {"classical (nu=0.3)":>19}')
    elastic = [f for f in shapes.frequency if f > 0.0]
    for k, frequency in enumerate(elastic):
        book = (f'{classical[k]:>19.3f}' if k < len(classical)
                else f'{"—":>19}')
        print(f'{frequency:9.1f}  {frequency * scale:9.3f}  {book}')