In the area of plate bending and displacement, Timoshenko and Krieger [1], remains a veritable source of reference including the contribution of many other expert-authors. The closed-form results of these experts are approximately but closely re-established by the numerical finite-element process since its inception about 1965; plate-stiffness reduces as loaded surface area increases, it was established. In the study of buckling here, a new fast but competitive displacement solution, based on domain spring-stiffness, is found and described.

Euler’s Eigen-vector ( Z sin π x / L ) for his original bar was not developed into a displacement-factor, (Δ). Buckling discovery quickly found applications in stability in frames of construction; in plates making up ships and air-planes, etc. With Kirchhoff-Love displacement-theory on the scene, the attendant Euler’s buckling displacement-factor should have been found given Euler mode ( W = Z sin π x / L ).

H x x = ∂ 4 w ∂ x 4 = ∂ 4 w / ∂ x 4 1 = ∬ w ( ∂ 4 w / ∂ x 4 ) ∂ x ∂ y ∬ w ∂ x ∂ y (1)

Find H x x = Z ( 1 / 4 ) π 5 / L 4 , H x x = 76.4977 Z , and, Δ i = Z W s h a p e = 0.013077 q * L 4 / D * -maximum.

There is nothing to separate this result from the Euler-Bernoulli beam displacement, −5/384 = 0.01302

A work-energy solution comes for comparison, not that it is essential. The reactive-capacity of the plate is first determined in terms of normal and twisting reactions for virtual continuous-loading.

The treatise of Arthur, W. Leissa, [2], 1985, gives a comprehensive discourse in buckling, encompassing shear-buckling. The complexities in buckling solutions quickly become apparent in the very first problem in “SSSS” where the square-plate and another “1. by 2.” solve exactly the same value of “N_cr ≡ 4” in uni-axial compression. Studies by Yaghoobi [3] found that a “1 by 1.5” SSSS-plate was 91-percent as strong as square plate in buckling. Experiments in buckling [4] [5] [6] show the same trend. The expected buckling-strength versus aspect-ratio of the “SSSS” plate in uni-axial compression is expected to be “Graph-B” rather than the stronger existing graph-A. The difference is important and has implications in many other situations including pure-shear. Already shown, Euler beam-buckling “ E I ⋅ ∂ 4 w / ∂ x 4 = F ⋅ ∂ 2 w / ∂ x 2 ” comes with a unique deflection factor of “ 0.013077 L x 4 / ( E I ) ” in a fundamental Sine-mode, close to the “0.01302 L⁴/EI” for “Euler-Bernoulli” beam in a polynomial statics mode; they play different roles. The buckling load of the “SSSS-plate” cannot have the same result between the aspect-ratios of 1.0 and “∞” as exhibited in existing monographs, as handed down since [1] [2]. The present study explains why it is untenable.

In Johnarry and Ebitei, [7] [8], buckling was tracked by displacement and the Euler-type Curve-B in Figure 1, was achieved; the buckling strength of infinitely long “SSSS-plate” reduced significantly below the “4.0” literature value, [1] [2]. Similar improvements were found in other cases and especially the pure-shear cases [8] [9] [10].

Pure-shear must always be investigated in plates, webs, and beam-column joints; indeed at all supports. The studies of Johns [9], Rockey [10] give great insights. This new spring-capacity solution of the “SSSS-plate” is carried over to the pure-shear case

and it was found the existing solution, “ N x y ≡ 5.351 + 4 / ( 1 + ( a / b ) 2 ) ” was too high [9], vis-à-vis a new solution [8], “ N x y ≡ 3.65 + 4 / ( 1 + ( a / b ) 2 ) ”. The webs of

beams in service are highly affected by pure-shear effects leading to wrinkling/buckling tending to change an existing shape; Figure 2. Pure shear always accompany heavy loading/reaction zones; buckling and pure-shear are composite in nature. It will be shown.

Vibration of plates continues to be an on-going study; unforeseen alarming vibration in cable-stayed bridges and recently in the London millennium foot-bridge (Year-2000) are recent cases. This is despite major design consideration of vibration in such cable-assisted long spans learning from earlier historic failures. Key aspects of early studies in vibration was due to Lord Rayleigh, [11], 1945, who confirmed that the free-vibration-factor of a simply-supported plate was “ ω 2 = 4 π 4 D / M ”. Many earlier results, up to 1966, were compiled by Bhalchandra Ballal, [12] showing that the then best all-clamped square-plate (CCCC) result was “ω ≡ 35.98” by “Iguchi” which is now known to be some 2.5% upper-bound. More recent extensive study and information on plates in free vibration are found in Authur M. Leissa [13] and in Xing and Liu, 2008 [14] who elaborated Levy-type deflection functions into characteristic equations and determinate de-composition for exact solutions; the lower-bound clamped frequency-factor for the “CCCC” was found [14] to be “ω ≡ 35.112” (lowest ever found). Adopting Navier-type conventional plate-bending analysis procedure should also yield comparable vibration results. Other simpler solutions are easily found equating potential and kinetic energies after Rayleigh-Ritz; or comparing

of energy-work potentials in a Galerkin solution. Onwuka, et al., [15] used such a technique with results, only 2.47-percent higher than Xi and Liu. [14] in frequency-factor, “ω” in the “CCCC”. So there had been good solutions since before 1966; what is essential now seems to be simpler self-checking design-oriented method as found in this new study through calibration from the exactly known all-sides simply supported plate.

The study of natural frequencies of materials and structures take pre-eminent stage because of effects of dynamic forces. The British Standard-5400, [16] prescribes dynamic forces in foot bridges as moving point-load of “ F ( t ) = 18 0 sine ( ω ⋅ t ) ” Newtons, simplifying dynamic designs. Similar suggestions can be made in other areas; the force “F(t)” can be an equivalent “Category-5 hurricane”.

The finite-element method basically comes down to “Stiffness-displacement-Load” [K]{Δ} = {F} equalit; by comparison, the Kirchhoff-Love plate equation for beam (w = Zf(x)) leads to Equation (1), where, [K]{Z} = {q_ef}. Both springs are basically the same; the former is, simply, more elaborate. The {F}, takes different forms for bending, buckling, vibration. In the particular case of buckling it is bending-moment, F ≡ {M_i}; P_cr/N_euler = (M_relative/Δ)/N_euler ≡ “moyope” in Table 1. In Ref. [17] with extensive computer buckling solutions print-outs, it was proved that finite-element beam-buckling followed “Sine variation”. For example the finite-element solution started as though the Sine-variation was not known and the initial loading was uniform, {q}; then replace {q} by the found {M} and keep replacing by new {M} until a steady state situation is found in three or four iterations. The closed-form solutions were, very accurately, re-established. The “FE” analysis in Table 1 for fixed-base sway-portal (I_beam/I_col = 0.1), buckling analysis is output-N_cr,col = 0.252N_euler. This is virtually the buckling of cantilever-column. (notice the steady-state solution). Tbe portal frame case is an extended beam analysis and only the beam-column finite-element is tested. By calling “I_b/I_c” = ∞(say, 100); “N_cr = N_euler”, and so on For beam from point-1 to point-2. length-L, the local stiffness elements are found. The matching Δ-vector reads {u, Δ₁, θ₁, Δ₂, θ₂] and the relevant force-action vector reads, {N, V₁, M₁, V₂, M₂}.

The logical sequence is, “[K_local] → [Transform] → [K_global]; explaining [K_local] as the basic slope-deflection, Table 2, with, r = EI/L; a^* = AE/L”; C₂₂ = 12 r/L²; and C₂₃ = −6 r/L; C₂₅ = 6 r/L.

In the “SSSS” plate, among others, buckling Eigen-values are unable to differentiate a square plate from a long one to the detriment of design. It will now be possible to specify buckling strength with characteristic displacement-factor to overcome this problem. It has already been shown the first Euler bar should have specified the characteristic-displacement, “ Δ = 0.013077 q * L 4 / E I ” from the solution Eigen-vector; it should have been the practice.

Uni-axial strength versus aspect-ratio buckling curves handed-down by experts, [1] [2] are connected individual mode-curves with unknown junction-values except by extrapolation; the present study will find continuous variation.

It will be shown that characteristic buckling displacement exists, (akin, to the Southwell-plot). Such relative-displacements find near-exact vibration-factors through calibration from the exact “SSSS” values.

∂ 4 w ∂ x 4 + 2 ∂ 4 w ∂ x 2 ∂ y 2 + ∂ 4 w ∂ y 4 = q D (2)

By expressing the deflection-“W” and the load “q” into similar infinite series Timoshenko and others have comprehensively explained the solutions for displacements and bending moments.

(a) Bi-axial:

∂ 4 w ∂ x 4 + 2 ∂ 4 w ∂ x 2 ∂ y 2 + ∂ 4 w ∂ y 4 = N x ( ∂ 2 w / ∂ x 2 ) + N y ( ∂ 2 w / ∂ y 2 ) D (3)

(b) Uni-axial in X:

∂ 4 w ∂ x 4 + 2 ∂ 4 w ∂ x 2 ∂ y 2 + ∂ 4 w ∂ y 4 = N x ( ∂ 2 w / ∂ x 2 ) D (4)

Many solutions of this are found in Ref. [2] and elsewhere but some are untenable, especially when the compressed ends are simply-supported.

(c) Pure-Shear:

∂ 4 w ∂ x 4 + 2 ∂ 4 w ∂ x 2 ∂ y 2 + ∂ 4 w ∂ y 4 = N x y ( 2 ∂ 2 w / ∂ x ∂ y ) D (5a)

Pure-shear solutions are not as available as the earlier two.

The method employed, here, is to relate the critical curvature in uni-axial compression to the von Misses distortion shear, (σ_VM), and associated curvature, “χ_VM.”

∂ 4 w ∂ x 4 + 2 ∂ 4 w ∂ x 2 ∂ y 2 + ∂ 4 w ∂ y 4 = N x y χ V M D (5b)

2 σ V M 2 = [ σ 1 − σ 2 ] 2 + [ σ 2 − σ 3 ] 2 + [ σ 3 − σ 1 ] 2 (5c)

Or, in respect of curvature, (χ)

σ V M = C ⋅ χ V M (6)

By Equations (5, 6) pure-shear is known with uni-axial buckling curvatures, [8].

∂ 4 w ∂ x 4 + 2 ∂ 4 w ∂ x 2 ∂ y 2 + ∂ 4 w ∂ y 4 = M ( ∂ 2 w / ∂ t 2 ) ( W sin ω ⋅ t ) D (7)

Or, with respect to dimension “x, y” only, for the eventual vibration equation,

∂ 4 w ∂ x 4 + 2 ∂ 4 w ∂ x 2 ∂ y 2 + ∂ 4 w ∂ y 4 = M ( ω 2 ) ⋅ W m o d e D (8)

To tie buckling and vibration, the next equation, Equation (9a), ensues; find [K_vib],

K b u c Δ b u c = K v i b ( Δ b u c + ∑ Δ h a m o n i c s ) (9a)

ω 2 = K v i b M (9b)

If buckling or elastic displacements were known, an alternate easier method using the “SSSS” as calibrator is,

ω n e x t 2 ⋅ Δ n e x t = ω s s s s 2 ⋅ Δ s s s s (10a)

Or, simply for a constant plate-thickness, even for the same boundary conditions,

ω 1 2 ⋅ Δ 1 = ω 2 2 ⋅ Δ 2 (10b)

The implication is that if the (ω²) is known for one aspect-ratio the value for another aspect-ratio with known “(Δ₂)” is readily found. For example, for the “SSSS” with “a/b = 1” and ω 1 2 = 389 . 6 and “Δ₁” = 0.00416 then at “a/b” = 2 with “Δ₂” = 0.0106, then, ω 2 2 = 152.9 , or, ω₂ = 12.36; that is all; there is no need to solve, afresh, the case, “a/b” = 2 (see Table 3, Table 4).

Relating the “SSSS” to “CCCC” for any aspect-ratio, the formula applies; for example, CCCC at a/b = 1: Δ c c c c → 0.00128 , then ω c c c c 2 → ( 0.00416 / 0.00128 ) × 389.6 = 1266 , ω c c c c = 35.58 … a result very difficult to improve upon, [12] [13] [14] [15].

The deflection function, w, is taken as

w = Z [ w s h p , x ] [ w s h p , y ] (19a)

The “SSCS” is considered in X-hydrostatic load q = q₀(x/a); Figure 3.

Bending was, initially, not the focus of the present study; this has been better treated elsewhere [1]. The relative deflections, as found here, needed for the studies in buckling, pure-shear, vibration are seen to be virtually, interchangeable with the bending ones.

(a): “SSCS”-Plate, Figure 3

1) w = Z [ sin G ⋅ x / a + A ⋅ x / a ] [ sin n π y / b ] (19b)

For n = 1, G = 4.5 and “A” = 0.977.

2) w * = ∬ w ∂ x ∂ y → 0.482 (19c)

3) H x x = ∬ w ( ∂ 4 w ∂ x 4 ) ∂ x ∂ y w ∗ (20)

yielding, → “ 0.225 G 4 / a 4 / w * → 191.42 ”.

4) H x y = ∬ 2 w ( ∂ 4 w ∂ x 2 ∂ y 2 ) ∂ x ∂ y w ∗ (21)

yielding, → “ 0.45 G 2 π 2 / w * → 186.6 ”.

5) H y y = ∬ w ( ∂ 4 w ∂ y 4 ) ∂ x ∂ y w ∗ (22)

yielding, → “ ( 0.384 / b 4 ) n 4 π 4 / w * → 77.6 ”.

Summing, H : ( H x x + H x y + H y y ) = 455.6 ]; noting, this spring-value will solve any loading for this plate.

Find “ Δ c r = ( 1 / H ) W s h p q n o r m ”.

Find (q_norm) as [ q i ⋅ w i w i ] by quadrature if not constant-q₀.

from points (0.5, 0.75); (0.25. 0.5); (0.5, 0.5); (0.75, 0.5); (0.5, 0.25),

find, (q_norm) as (0.466q₀) and so, Δ c r → ( 1 / H ) × 1.266 × 1.0 × 0.466 → 0.001295 ; cf, 0.0013, [1].

The solution is in the context of the “finite-element-method: [K]{Δ} = F. In this case [K or H] = 455.6.

(b): “udl case”: If “q_norm” ≡ q₀; udl, Δ_cr → 0.001295/(0.466) → 0.002779; cf, 0.0028 [1]. The new method is, therefore, self-checking.

Three possibilities are identified relative to X- and Y-axes in emulating the reactive potentials “[∂⁴w/ ∂x⁴]; [2∂⁴w/∂x²∂y²]; [∂⁴w/∂y⁴]”. No curvature “ X i ” is ever applied in practice but “ X x ” and “ X y ” act jointly.

1) σ x X x

This is first in contention in uni-axial X-compression; this case easily solves Equations (3, 4).

2) σ y X y

This is out of contention when no load is applied in the Y-axis, whatever the value of “ X y .”

3) σ x X a v

This “average loading-curvature” situation will always happen and also in contention. 1) and 2) are identified in the Mohr’s diagram, Figure 4.

In effect, two curvature-loading circles ( X x , X a v ) are operative and the larger circle gives the required solution for “N_x”. This process softens the stiff constraint that the wave numbers “m, n”, must be whole numbers.

(a). Importance of Curvature, “(𝜕²w/∂x²)_cr”, in Uni-axial-Compression Buckling

The solution of Equation (3). is easy when

1): when, “ ∂ 2 w / ∂ x 2 ≥ ∂ 2 w / ∂ y 2 ”

or

2) when, X x = X c r = ∂ 2 w / ∂ x 2

When “ ∂ 2 w / ∂ x 2 < ∂ 2 w / ∂ y 2 ”, “ X 1 ” may be interpreted as principal loading curvature that must align with “(∂²w/∂x²)” approximately or else find (χ) by equation (20).

So, Figure 4, explaining Mohr’s loading curvatures will supply the critical solution curvature.

(b)-Alternate Critical X-Curvature, “(𝜕²w/∂x²)_cr” By Deflection Ratio

Relying on the deflection coefficients, Δ₁, Δ₂ at two consecutive locations, “i” and “i + 1”, Figure 5(a), the curvature at the second location may be found from Equation (17)

Δ 1 X x , 1 A 1 = Δ 2 X x , 2 A 2 (23a)

A₂/A₁ = C_A, stressed boundary lengths-ratio representing side areas; Figure 5(b), Figure 5(c).

Aspect ratio, “s^*” gap of 25-percent can be tolerated.

This equation is similar to Equation (3) as “(Force/Area)(curvature) = Constant” = Kirchhoff’s plate-capacity.

A = 2 b + 2 a t 3 (23b)

From Equation (4), put, N x ( ∂ 2 w / ∂ x 2 ) ≡ q * ; for a given plate the “LHS” is invariant and once computed can be used for bending, buckling and vibration; “q^*” is equivalent uniform transverse pressure. That is, Δ ≡ ( 1 / H ) w s h a p e ; the primitive buckling-value is sufficient. The familiarity of “Δ value” gives confidence the solution is on track.

(a); a/b = 0.95; Deflection in Buckling

w = Z [ sin G ⋅ x / a + A ⋅ x / a ] [ 1 − cos n π y / b ] (24)

in which, “n = 2;G = 4.5; A = 0.977; b = 1”.

By Equation (19), find, w^* = 0.757.

By Equation (20), find “H_xx” as → ( 0.675 G 4 / a 4 ) / w * → 365.74 / a 4

By Equation(21) find “H_xy” as → 0. 45 n 2 ⋅ π 2 ⋅ G 2 / a 2 ⋅ b 2 → 475 . 24 / a 2

By Equation (22), find “H_yy” as → ( 0.383 n 4 ⋅ π 4 / b 4 ) / w * → 788.46

Find curvatures,

χ x = ∬ w ( ∂ 2 w / ∂ x 2 ) ∂ x ∂ y w ∗ (25)

yielding, → “ ( 0.675 K 2 / a 2 ) / w * → 18.056 / a 2 ”.

χ y = ∬ w ( ∂ 2 w / ∂ y 2 ) ∂ x ∂ y w ∗ (26)

yielding, → “ ( 0.383 n 2 ⋅ π 2 / b 2 ) / w * ≅ 20.03 / b 2 ”.

Evaluating, a/b = 0.95; [ X x = 18.06 / 0.95 2 = 20.0 ]; [ X y = 20.03 ≡ 20 ]; [H_xx = 448.49; H_xy = 526.58; H_yy = 788.46; H = 1763.53];

Δ s * = 0.95 = W s h p - c / H → Δ c r ; find W_shp → (w_shp,x)(w_shp,y); at x/a = 0.4 and y/b = 0.5, w_shp → 2.73; (compare the shape-value of “CCCC”-plate at center of bending → “w_shp = 4”, from function. W ≡ Z ( 1 − cos 2 π y / a ) ( 1 − cos 2 π y / b ) . So, Δ_cr → 0.00155.

Curvature and Buckling Value, (a/b = 0.95), N_x,cr;

→ N x , s * = 0.95 = H / X x , c r → 8.93 ρ (expected to be exact).

(b) Try New Aspect-Ratio, a/b = 1; the “SCCC”; H = 1627; χ_x = 18.88 (with help of Equation (23))

N x , s * = 1 → 1627 / 18.88 → 8.73 π 2 ; cf. 8.986, [2]; table is skipped similar to Table 6.

(a) The “CCCC” buckling-plate; “Δ-method”; Table 3

w = Z [ 1 − cos m π x a ] [ 1 − cos n π y b ] (27)

in which,; m, n = 2, …

(i): a/b = 1; By Equations (20, 21, 22, 25, 26), H_xx = 1168.8; H_xy = 779.2; X x = X y = X c r = 29.608 ; H_yy = 1168.8; H = 3116.8; N s * = 1 = 31168.8 / 29.608 = 10.66 ρ ; Δ_c = 0.00128; cf, 0.00126 [1]; the nearness of the primitive-Δ to the final-Δ, (0.00128 to 0.00126), confirms “H” on which “N_x” depends. Further, the exactness of the w-function is verified.

(b) More Results: More results are found in Table 3

The “CCCC” is sampled in Table 3 in the “Δ-method” to reflect aspect-ratios; for “ X i ” from “ X i − 1 ”.

(Δ-method), is compiled in Table 4, for “w = Z(sinπx/a)(sinπy/b)”.

w = Z [ 1 − cos m π x a ] [ sin π x / a ] (28)

where m = 2; results are displayed in Table 5; at s^* = 1 the Δ-factor, 0.00194, are very accurate in the same fashion as the “CCCC” plate in Table 3. The results for pure-shear match the literature “SSSS” values and so the “SSSS” literature-values are untenable. The new “SSSS” pure-shear values as given in Table 4 are only about 60% of those of ‘CSCS’ in Table 5.

w = Z [ 1 − cos m π x a ] [ sin ( G ⋅ y b ) + A ⋅ y / b ] (29)

in which, m = 2;G = 4.5, A = 0.977; proceeding as before sample results are shown, (Table 6).

where for “s^* = 1” the buckling load is N_x = 8.2ρ and the vibration-factor, ω_cscc, is 31.97 (cf, 31.44 [12]; these are near exact results. The discrepancy in “ω” is truncation error that can become smaller as the X-wave number, m, goes beyond “10”. Results are highly accurate in displacement and buckling factors.

w = Z sin π x a [ 1 − cos 2 π y b ] (30)

Find that the aspect ratio, s^* = 0.866 is critical where “ χ_x = χ_y “.

Some results are sampled in Table 7; notice that “ N c r , s ∗ = 1 is exact at 7.79” All displayed are very accurate.

In the Tables 3-6, the factor, ( δ X / δ Δ ), exhibits a critical stationary-point, Figure 7, where “Δ_limit” is sampled.

The minimum buckling load, for “very-long” plates, is indicated at that point, whatever the values of “m, n or s^*”. The relative weakness of a plate is indicated in bending-buckling-vibration-analyses [1] [17]. By Yaghoobi [3] the buckling strength of the “SSSS” at s^* = 1.5 is 91-percent of the value at s^* = 1; this is the kind of statement sought-after here. This sits well with the ratio of 89-percent in Table 3; also experiments [6], showed a “s^* = 2 plate was only some 40-percent as strong as a “s^* = 1” one. In “plate buckling solution based on characteristic deflection-factors”, [7], relevance of deflection was focused on. Here, the analysis starts with beam-strip solutions that are already fully known; for simply supported strip, Δ_c = 5/384 = 0.01302 and the “SSSS” plate ends in this value when it is very long or very short; for example check the Δ_c = 0.0135 in Table 3 in the “SSSS” at s^* = 3.5; so the size of “Δ” can, also, be used to terminate solutions.

Once the solution of the “SSSS” plate is found, it is only necessary to confirm the buckling-vibratory deflection, (Δ_next) of any next-plate in any other support configuration so as to emulate the “SSSS”. By Equation (9),

M n e w ω n e w 2 Δ n e w = M s s s s ω s s s s 2 Δ s s s s (31)

Equation (31) is profoundly important; it implies that if in the process of vibration a boundary condition reduces from a higher to a lower one, the plate will not collapse provided there is no significant serviceability damage. A “CCCC” plate reducing to “SCCC” only brings higher amplitude until inspection and retrofitting.

The buckling deflection is recommended because [K] is already known, at this stage, and from Equation (19b), “ Δ = Z w s h p , x ⋅ w s h p , y ”.

Z = q * H (32a)

For example, find the ( ω c c c c 2 ) from ( ω s s s s 2 ) as follows,

From Table 3, “ Δ b u c k , c c c c = 0.00128 ”; So,

“ ω c c c c 2 → 389.635 × 0.00416 × ( 1 / 0.00128 ) = 1266.3 ; so, ω c c c c = 35.585 ; cf, 35.112, Xing [14]; 35.8, [13] ”

Using Timoshenko and Kreiger’s, [1] deflection factors make very little difference, that is

“ ω c c c c 2 = 389.6 × 0.00406 × 1 / 0.00126 = 1255.38 ; ω c c c c = 35.43 .”

It is now easy to build up Table 8 covering many plates; extendable; these results are usually more exact than those from independent computations from, “ [ K ] { Δ } = ω 2 [ M ] { Δ } ”

The method easily detects error entries; in Xing and Liu’s [14] the “CSCC/2” plate as “ω = 24.1” was in error; the correct result is “ω = 18.2”; the difference is large. The plate “CCCC/2” which has “ω_cccc/2 = 24.36” is much stiffer than “CSCC/2”, by their deflection results. So, there is near-perfect match for Xing and Liu’s results with the “SSSS” as calibrator.

This may be achieved by invoking Equation (9). Since buckling attributes are already listed in the Tables, it is only necessary to compute some additional sympathetic harmonics to evaluate the results. Already found results and new ones will be established easily. Indeed, Equation (9a) led to the “CCCC” independent results in Table 3.

(a) Practical Sampling of Vibration-Harmonics

A “3,1” mode in the “SSSS” means three half-sine waves. The equations may be re-visited to find “(ω_ssss,3,1)².”

Test the “CCCC” for (ω_cccc,2,1); a/b = 1. This is not an important harmonic as it does not amplify the fundamental as shown in Figure 8. The “CCCC” curve in the X-direction splits into two modes of “CCSC” each;

using existing principle to analyse one “CCSC” for “a = 1/2 and b = 1”.

“{With the mode, Q = ( sin 4.5 ( x / a ) + 0.977 x / a ) ( 1 − cos 2 π y / b ) ; find [ H x x → 0.89 G 4 / a 4 = 5839 ; H x y = 0.59 n 2 G 2 π 2 / ( a 2 b 2 ) = 1886.7 ; H y y = 0.507 n 4 π 4 / b 4 = 790.1 ; H = 8516 ]

Z = 1 / H = 0.000117 ; Δ = Z W s h p ; W s h p = 1.34 × 2.0 at “x = 0.35, y = 0.5”;; So, Δ_2,1 = 0.000315”}.

Finally, calibrating from the “SSSS”.

ω c c c c , 2 , 1 = 389.6 × 0.00416 / 0.000315 (32b)

→ “71.73 radians; cf, 72.889” [14].

For very speedy processing, Equation (28) for the “SSSS” is modified for other plates by interpreting the lengths “a, b” as effective Euler-length-averages between waves; some refinement will be possible (Figure 9).

ω 2 = π 4 D M [ J 2 ] (33a)

J = 1 a 2 + 1 b 2 (33b)

ω 2 = π 4 D M [ S 2 ] (34a)

where,

S = 1 L e - a v - x 2 + 1 L e - a v - y 2 (34b)

For simplicity, the approximation in Equation (34) for the harmonic (≥3) is employed (mode = number of peaks)

L e - a v = 1 modenumber . (34c)

Results improve fast as harmonics increase beyond “2”.

Some typical results are listed in Table 9 below.