Natural frequencies of plates

A plate vibrates freely at its natural frequencies, each with a shape of its own, the mode shape. Near one of them, a small periodic force, such as from a motor, a fan or a vehicle, makes the plate vibrate strongly in resonance, so a design keeps the natural frequencies clear of the excitation frequencies. Formula collections give the frequencies of rectangular and circular plates with standard supports. For any other outline, with holes and supports on parts of its edges, they follow from the eigenvalue problem of the plate's equations of motion. The tool meshes the plate, solves that problem with the finite element method, which is a modal analysis, and shows each mode with its frequency and its nodal lines.

Chladni figures

Ernst Chladni strewed sand on metal plates and bowed their edges. At a natural frequency, the plate vibrates in one of its modes, and the sand gathers on the nodal lines, where the plate stands still. The tool draws each mode's deflection in color, red on one side and blue on the other, with the nodal lines dark where the deflection changes its sign. With Sand on, it shows the plate dark and the sand along the nodal lines instead, where the deflection is small. A mode that moves to one side everywhere has no nodal line, such as the third mode of the Chladni plate sample, which is clamped at its middle.

The plate model

The plate is a Reissner-Mindlin plate, whose deflection w and rotations are unknowns of their own, coupled by the shear. For a thin plate, the model gives the frequencies of Kirchhoff's classic thin plate theory, and for a thicker one, it includes the shear deformation and the rotary inertia, which lower the higher frequencies. The material enters through Young's modulus E, Poisson's ratio ν and the density ρ, and the thickness t through the bending stiffness D = E t³ / (12 (1 − ν²)). For a thin plate, every frequency grows in proportion to the thickness.

Edge and hole numbers

Every edge of the outline gets a number, from E1 at the lowest left corner around the plate, and every hole one number for its whole boundary, H1, H2 and so on from the left. The supports refer to these numbers, and the drawing shows them, with each edge's length and each hole's size on hover. The Propagation column carries a support on along the outline, with planex2d's selection propagation.

• Minimal keeps the edge alone.
• Coincident adds the edges joined to it end to end, which on a closed outline is all of it.
• Tangent adds the edges that continue it tangentially, such as the lines on either side of a fillet.
• Curvature adds the edges that continue its curvature as well.

Supports

A clamped edge or hole can neither move nor turn. A simply supported one cannot move out of the plane but can turn about the edge, like a plate resting on a knife edge. Edges and holes not in the table are free. A clamped hole stands for a screw or a post that holds the plate. The supports are edges and holes, not single points, so a standoff is a small hole drawn at its place and clamped. A free plate has three rigid-body modes at zero frequency, in which it moves up and down or tilts as a whole without bending, and the tool leaves them out.

Results

The results are the frequencies of the lowest modes, eight by default, the plate's mass and the mesh. A click on a mode shows it in the drawing, and its play button sounds its frequency for a moment, for frequencies from 20 Hz to 20 kHz. The deflection of each mode is scaled so that its largest value is 1, since a mode gives the shape of the vibration and not its size. The nodal lines of the mode shown can be saved as a DXF or SVG file, with the plate's outline on a layer of its own, for example to engrave the figure. The PDF report draws the modes in the look the Sand switch sets.

KiCad boards

A KiCad board (.kicad_pcb, KiCad 9 and later) gives its outline on Edge.Cuts and the holes through the board. Holes under 2 mm across, such as vias and the holes of component pins, are left out, since they change the frequencies little and would need a very fine mesh. Components add mass and some stiffness, which usually lowers the frequencies of a populated board below those of the bare board the tool computes.

Simplify outline

Drawings often carry arcs and holes as chains of short lines, which give the outline many edge numbers. With Simplify outline on, planex2d fits arcs to such chains, within a ten-thousandth of the plate's size, before the plate is numbered and meshed. Turn it off to analyze the outline exactly as drawn. Where the fit misses an arc, the contour checker fits arcs with a tolerance and an edge length of your choice, closes gaps and removes tiny edges, and saves the cleaned drawing for the plate tool.

Units

Lengths are in millimeters, Young's modulus in GPa, the density in g/cm³ and frequencies in hertz. A file is read in the unit it names and converted to millimeters.

1. Load

   The page loads Python and planex2d into this tab, and NGSolve with the first solve, about 23 MB the first time. Your browser keeps them in its cache, so later visits start faster.

2. Read

   planex2d reads the file, joins its closed contours into the plate, simplifies the outline and numbers its edges and holes.

3. Solve

   NGSolve meshes the plate and finds its lowest modes, again whenever a setting changes after the first solve.

4. Save

   The modes are saved as a PDF report or a CSV table, and the nodal lines as a DXF or SVG file, made in the tab as well.

The finite element model

The mesh has elements of about a twelfth of the plate's size, finer along arcs and holes, and it is curved to the exact outline. The deflection uses elements of order 4 and the rotations elements of order 3, one order lower. The Solver group sets the deflection's order from 2 to 6. Knyazev's LOBPCG eigenvalue solver finds the lowest modes on NGSolve's matrices, preconditioned with the sparse Cholesky factorization of the stiffness plus a small part of the mass. The tests compare the tool with known answers. A free square plate gives the tabulated frequency coefficients within 0.03 %, a simply supported rectangle the closed form (π/2)((m/a)² + (n/b)²)√(D/(ρt)) within 0.01 %, and a clamped circle its coefficients within 0.03 %.

Supported files

Run it in Python

The same analysis takes a few lines with planex2d and NGSolve. The example is a free steel plate of 240 × 240 × 1 mm, held nowhere, as in the Chladni figures example of the docs. planex2d draws the plate, NGSolve meshes it and finds its lowest modes with NGSolve's LOBPCG, which uses SciPy, and the first three, the rigid-body modes, are left out. See Chladni figures in the docs for the plate's nodal lines.

$ pip install "planex2d[occ]" ngsolve scipy

import math

import ngsolve as ngs
import planex as px

a, h, E, nu, rho = 240.0, 1.0, 210e3, 0.3, 7.85e-9
D, G = E * h**3 / (12 * (1 - nu**2)), E / (2 * (1 + nu))
plate = px.QuickSketch().RectangleC(a, a).Face()
mesh = ngs.Mesh(px.OCCGeometry(plate).GenerateMesh(maxh=a / 12))
mesh.Curve(4)
X = ngs.H1(mesh, order=4) * ngs.VectorH1(mesh, order=3)
(w, th), (v, et) = X.TnT()
bend = ngs.Sym(ngs.Grad(th)), ngs.Sym(ngs.Grad(et))
stiff = D * ((1 - nu) * ngs.InnerProduct(*bend) + nu * ngs.div(th) * ngs.div(et)) \
    + 5 / 6 * G * h * (ngs.grad(w) - th) * (ngs.grad(v) - et)
mass = rho * h * w * v + rho * h**3 / 12 * th * et
K = ngs.BilinearForm(stiff * ngs.dx).Assemble()
M = ngs.BilinearForm(mass * ngs.dx).Assemble()
S = ngs.BilinearForm((stiff + D / (rho * h * a**4) * mass) * ngs.dx).Assemble()
pre = S.mat.Inverse(X.FreeDofs(), inverse="sparsecholesky")
lams, _ = ngs.solvers.LOBPCG(K.mat, M.mat, pre, num=8, maxit=20, printrates=False)
for lam in sorted(lams)[3:6]:
    print("frequency", math.sqrt(lam) / (2 * math.pi), "Hz")

It prints the frequencies 58.22, 84.74 and 104.95 Hz, against 58.22, 84.75 and 104.96 Hz from the tabulated coefficients 13.463, 19.596 and 24.270 of the free square plate.

Privacy

Limits

• The tool computes the free vibration of a linear elastic plate, with small deflections and no damping. It gives the frequencies and mode shapes, not how far the plate moves under a given excitation.
• The material is isotropic. FR-4 is close to that in its plane, while wood and other fiber materials are much stiffer along their fibers than across them.
• Components on a board, added masses, stiffeners and the air around the plate are not modeled.
• The plate vibrates out of its plane only. Bending under a static load is not part of the tool.
• The tool analyzes one connected plate. A file must be planar, and its text, dimensions and hatches are not read.
• A plate that fails or a result that looks wrong? Open an issue on GitHub and attach the file if you can.