Plate Vibration

Plate Vibration Method

Plate Vibration: Exact, Lévy and Rayleigh-Ritz Solutions, Warburton's Formula and the Circular-Plate Bessel Equations

What the tool computes

The natural frequencies and mode shapes of a thin, flat, isotropic plate vibrating freely in bending: a rectangular plate with any combination of simply supported, clamped and free edges, or a solid circular plate with a clamped, simply supported or free edge. It follows Leissa's compilation Vibration of Plates (NASA SP-160, 1969), which collected every solution then known and tabulated the frequency parameters, and it prints Leissa's table values beside the computed ones wherever his tables cover the case. For the rectangular plate it also reports Warburton's closed-form estimate for every mode and the shift of the frequencies under uniform in-plane forces. The page draws the first three mode shapes and plots the frequencies against the mode number.

Conventions

Rectangular plate 0 ≤ x ≤ a, 0 ≤ y ≤ b; edges 1 (x = 0), 2 (x = a), 3 (y = 0), 4 (y = b). Leissa writes the edge conditions in the order x = 0, y = 0, x = a, y = b (so SS-C-SS-F is simply supported at x = 0 and x = a, clamped at y = 0, free at y = b), and the page prints that code. The thickness is h, the density ρ, the mass per unit area ρh, and the flexural rigidity D = E h³/[12(1 − ν²)]. Leissa's dimensionless frequency parameter is λ = ω a² √(ρh/D) for the rectangular plate (referred to the side along x; several of his tables use b instead, and the page converts them with (a/b)²) and λ² = ω a² √(ρh/D) for the circular plate of radius a, where λ = ka is the root of the Bessel characteristic equation. The frequency in hertz is f = ω/2π. Mode indices (m, n) follow Warburton: the number of nodal lines along x and along y counting the boundaries but not free edges; for a simply supported plate they reduce to the numbers of half-waves. In-plane forces N_x, N_y are per unit width, tension positive.

The governing equation

Kirchhoff's thin-plate theory (straight normals, no transverse shear strain, no rotary inertia) gives for the free transverse vibration D∇⁴w + ρh ∂²w/∂t² = 0; with w = W(x, y) cos ωt this is ∇⁴W − k⁴W = 0, k⁴ = ρhω²/D (SP-160 eq. 1.4). Two conditions apply on each edge: simply supported W = 0, M_n = 0; clamped W = 0, ∂W/∂n = 0; free M_n = 0, V_n = 0 with Kirchhoff's effective shear V_n. Uniform in-plane forces add N_x ∂²w/∂x² + N_y ∂²w/∂y² to the left-hand side (SP-160 Chapter 10). The frequency parameters of plates without free edges do not depend on Poisson's ratio except through D; with a free edge they do, which is why Leissa's tables for such cases are stated at one value of ν and the page compares with them only at that value.

All edges simply supported: the exact solution

W = sin(mπx/a) sin(nπy/b) satisfies the equation and every boundary condition, giving ω_mn = π²[(m/a)² + (n/b)²] √(D/ρh), or λ_mn = π²[m² + n²(a/b)²] (SP-160 eq. 4.20). The square's fundamental is 2π² = 19.739. Under in-plane forces the same modes remain exact and ω²_mn = [D(α² + β²)² + N_xα² + N_yβ²]/ρh with α = mπ/a, β = nπ/b: the square of the frequency falls linearly with a compressive force and reaches zero at the buckling load of that mode, N_x,cr = −D(α² + β²)²/α²; the page reports the critical force of each mode and plots the fundamental against N_x.

Two opposite edges simply supported: the Lévy cases

With edges x = 0 and x = a simply supported, W = Y(y) sin(mπx/a) (Lévy's form) satisfies those two edges identically and reduces the problem to the ordinary equation Y'''' − 2α²Y'' + (α⁴ − k⁴)Y = 0, α = mπ/a (SP-160 eq. 4.25). Its solution combines sin λ₁y, cos λ₁y, sinh λ₂y, cosh λ₂y with λ₁² = k² − α² and λ₂² = k² + α² (eq. 4.27); the two conditions on each of the other edges give a 4 × 4 homogeneous system whose determinant must vanish, which is the characteristic equation: for SS-C-SS-C, 2λ₁λ₂(cos λ₁b cosh λ₂b − 1) + (λ₁² − λ₂²) sin λ₁b sinh λ₂b = 0 (eq. 4.29), for SS-C-SS-SS λ₂ cosh λ₂b sin λ₁b = λ₁ sinh λ₂b cos λ₁b (eq. 4.33), and the longer forms of eqs (4.41) and (4.45) for a free edge, which bring in ν. Voigt (1893) gave the first of these exact solutions; Iguchi (1938) computed the roots for the square that Leissa reprints.

The tool does not transcribe each printed equation; it builds the 4 × 4 boundary matrix numerically for any pair of conditions (clamped, simply supported or free at y = 0 and at y = b), on a basis chosen so that the determinant is an entire function of s = k² − α² (sin √s y/√s and cos √s y pass smoothly into their hyperbolic forms when s turns negative, and the hyperbolic pair is divided by cosh λ₂b so that nothing overflows). The roots are located by scanning s from −α² upward and refined by bisection, harmonic by harmonic, until the requested number of lowest frequencies is certain. This matters for SS-F-SS-F and SS-SS-SS-F plates, whose lowest modes have k² < α² (SP-160 Table 4.21 gives the condition m b/a < (√2/π)(2/ν − 1)², 14.455 for ν = 0.3); Leissa notes that one published table mishandled them. The mode shape is the null vector of the boundary matrix at the root. When edges 3 and 4 are the simply supported pair the tool swaps the axes internally and converts back. Every value agrees with Leissa's exact tables (4.3, 4.6, 4.8, 4.10, 4.13, 4.16, 4.18) to the printed digits, and with the roots of Leissa's printed equations found independently with scipy.

Other combinations: the Rayleigh-Ritz method with beam functions

For the remaining 15 of the 21 combinations no separable exact solution exists. Rayleigh's method takes the mode as a product X(x)Y(y) of the normal modes of beams having the plate's edge conditions, which satisfy all the geometric conditions and, except the shear condition of a free edge, the natural ones too; Ritz's extension (1909, for this very problem, the free square) expands the deflection in many such products, W = ΣΣ A_ij X_i(x)Y_j(y), and makes the Rayleigh quotient stationary. Young (1950) carried this out with 18 terms for the clamped square, the cantilever and the C-C-C-SS plate, Warburton (1954) with one term for all 21 cases, and Leissa (1973) with 36 terms for the whole set. The tool uses N functions in each direction (10 by default, 100 terms; 6, 8 or 12 may be chosen): the simply supported pair's sines; the clamped-clamped, clamped-free, clamped-pinned and pinned-free beam modes in the stable form pe^z + qe^−z + c₃ cos z + c₄ sin z (so that clamped-end values of the high modes do not suffer from cancellation of large hyperbolic terms); and for a free-free pair the rigid translation and rotation followed by the elastic modes. Each set is normalised and its integrals ∫X_iX_k, ∫X_i'X_k', ∫X_i''X_k'' and ∫X_i''X_k are evaluated by composite Gauss-Legendre quadrature (288 points), from which the strain energy D/2 ∫∫[w_xx² + w_yy² + 2ν w_xxw_yy + 2(1 − ν)w_xy²]dA, the in-plane work ½∫∫[N_xw_x² + N_yw_y²]dA and the kinetic energy ½ρhω²∫∫w²dA are assembled as Kronecker products. The generalized eigenproblem Kc = ω²Mc is solved completely (Cholesky reduction, Householder tridiagonalisation, implicit QL), the zero eigenvalues of rigid-body motion are dropped, and each mode is labelled by its dominant product (m, n).

The Rayleigh-Ritz frequencies are upper bounds that decrease as terms are added, and the page prints the same solution with two fewer terms per direction as a convergence record. With 10 terms the clamped and simply supported combinations are converged to about 0.01 % (the clamped square gives 35.986 against the exact 35.985 and Bazley, Fox and Stadter's 50-term bound 35.986); the free-edge cases converge more slowly because the beam functions cannot meet the shear condition, to roughly 0.5 % for one free edge and 1 % for several (the square cantilever 3.484 against Young's 18-term 3.494 and the free square's twisting mode 13.48 against Leissa's 36-term 13.49). The default plate size of 10 terms runs in about a second; 12 terms take a few seconds.

Warburton's formula

Warburton's one-term Rayleigh result (SP-160 eq. 4.16) is λ = π²√{G_x⁴ + G_y⁴(a/b)⁴ + 2(a/b)²[νH_xH_y + (1 − ν)J_xJ_y]}, with the constants G, H, J of Table 4.1 for each edge pair and mode index (for a simply supported pair G = H = J^½ = m − 1; for a clamped pair G = m − ½; for a clamped-free pair 0.597 and 1.494 for the first two modes; and so on). The page lists it for every computed mode. It is exact for the simply supported plate, within half a percent for clamped pairs, but 10 to 15 % high for torsion-like modes of plates with free edges, where a single product cannot represent the twisting surface; the comparison column shows where the one-term estimate can be trusted.

Circular plates

In polar coordinates the separated solution is W = [A J_n(kr) + C I_n(kr)] cos nθ (SP-160 eq. 2.1; the Bessel functions of the second kind are excluded by regularity at the centre), n being the number of nodal diameters and s the number of interior nodal circles. The edge conditions at r = a give, with λ = ka: clamped, J_n(λ)I_n'(λ) − I_n(λ)J_n'(λ) = 0, independent of ν (eq. 2.5; λ² = 10.2158, 21.26, 34.88, 39.771 …, Table 2.1); simply supported, J_n+1/J_n + I_n+1/I_n = 2λ/(1 − ν) (eq. 2.11); free, the vanishing of the determinant of M_r = 0 and V_r = 0, with V_r = Q_r − (1/r)∂M_rθ/∂θ, whose roots depend on ν (Tables 2.5 and 2.6 at ν = 0.33 and 0.25); the three rigid-body motions of the free plate (n = 0 translation and n = 1 tilt at λ = 0) are not modes. The tool evaluates J_n and e^−xI_n from their integral representations by the trapezoidal rule, which is spectrally accurate for these periodic integrands, scans λ for sign changes of each characteristic function and bisects to the root. Every n ≥ 1 mode is a degenerate pair (cos nθ and sin nθ) and is listed once.

Leissa's tables and what was found in them

The page prints Leissa's values beside the computed ones for every tabulated case: the exact Lévy tables, Young's and Ödman's clamped-plate values, Bazley, Fox and Stadter's bounds, Dill and Pister's, Kanazawa and Kawai's and Hamada's values for C-C-C-SS and C-C-SS-SS, Claassen and Thorne's for C-F-C-F, Young's and Barton's cantilevers, the free square at several Poisson's ratios, and the circular tables. Each printed value is matched with the nearest computed mode, because several tables list one symmetry class or a few mode families rather than the lowest frequencies in order (Barton's five cantilever "modes" include the chordwise-bending family, which for a long cantilever is far up the list). Reading the tables page by page also turned up a few entries that are not roots of the book's own equations: the simply supported circular plate's Table 2.3 runs 0.2 to 0.9 % above the roots of eq. (2.11) (4.977 printed against the exact 4.935 for the fundamental, which Prescott's 2.204 at ν = 0.25 on the same page confirms); Table 4.6 prints 24.047 for SS-C-SS-C at a/b = 1.5 where the exact root is 25.044 (a misprint, the row being smooth only with 25.047); and three entries of Table 4.17 (SS-SS-SS-F at a/b = 0.6, 0.8 and 5.0) lie 1 to 7 % above the exact roots. The tool prints the exact values in every such case and says so on the page; the test script records each discrepancy against an independent scipy evaluation.

Reading the results

The frequency table lists the modes lowest first with their indices, λ, ω and f; a closed form or the characteristic equation is written out for the exact cases and the convergence record for the Ritz cases. Degenerate pairs (the square's (1, 2) and (2, 1), every n ≥ 1 mode of the circular plate) appear as equal frequencies. The in-plane-force section gives the loaded frequencies with the critical force of each mode for the simply supported plate, and the Ritz solution with the geometric term otherwise; a mode whose frequency has reached zero has buckled. Figure 1 draws the first three mode shapes (red positive, blue negative, black nodal lines; the sign of a mode is arbitrary) with the edge conditions marked; Figure 2 plots the frequencies against the mode number with Warburton's estimates and the matched table values.

Limits

Thin-plate theory needs a thickness below about a tenth of the shorter span and mode wavelengths well above the thickness; beyond that, transverse shear deformation and rotary inertia (Mindlin) lower the frequencies, more for the higher modes, and the page warns. The plate is isotropic, of uniform thickness, without stiffeners, added masses, holes, elastic edge restraints or point supports, and the circular plate carries no in-plane force. The frequencies are those of free undamped vibration; the forced response, damping and the frequency content of a base excitation are the subject of the Random Vibration Response and Shock Response Spectrum tools. A free-free plate's rigid-body motions are removed from the mode list; with an in-plane force they acquire a membrane-like stiffness and reappear in the loaded list, which the page notes. Laminated plates belong to the Classical Laminate Theory tool and buckling under in-plane loads to the Plate Buckling tool.