.18Classical and FEM Solutionsof Plate Vibration18–1Chapter 18: CLASSICAL AND FEM SOLUTIONS OF PLATE VIBRATION18–2§18.1 INTRODUCTIONClassicalsolutionsofplatevibrationshavebeenasubjectofmanyeminentscientistsandengineersinthepast.A compendium by Leissa [1] is perhaps the most comprehensive source to date. As the simply supportedplates (SS-SS-SS-SS) are the simplest to address, we recall from the previous chapter their solution:Vibration mode shapes: W(x, y) = sinαx sin γ y,αma = πm, m = 1, 2,...;γma = πn, n = 1, 2,...Frequency equation: β4mn=ω2mnρD,β2mn= π2[(ma)2+ (nb)2]D =Eh312(1 − ν2)(18.1)where (m, n) denote the number of harmonics along the x and y-coordinate directions.In conformity with literature, we will modify the frequency equation asλmn= ωmna2ρD,λmn= (βmna)2(18.2)where a is the x-directional plate length. Hence, for a rectangular plate (a = b) we have the followingfrequency equation:λmn= π2(m2+ n2)(18.3)We now summarize classical solutions of rectangular plate vibration.§18.2 ASSUMED VIBRATION MODE SHAPES FOR SIX BOUNDARY CONDITIONSThe classical Rayleigh method assumes the plate deflections as the product of beam functionsW(x, y) = Y(y) X(x)(18.4)each of which can be chosen depending on the boundary conditions.§18.2.1 Simply supported at x = 0 and x = a (SS-SS):X(x) that satisfies the boundary conditionsW(y, 0) = W(y, a) =∂2W(y, 0)∂x2=∂2W(y, a)∂x2= 0 (18.5)can be expressed asX(x) = sin(m −1)π xa, m = 2, 3, 4,... (18.6)It should be noted that if SS boundary conditions are imposed along y = 0 and y = b, then (x, a, m) in theabove expression is simply replaced by (y, b, n).18–218–3 §18.2 ASSUMED VIBRATION MODE SHAPES FOR SIX BOUNDARY CONDITIONS§18.2.2 Clamped at x = 0 and x = a (C-C):For even harmonics: X(x) = cos γ1(xa−12) +sin(γ1/2)sinh(γ1/2)coshγ1(xa−12)tan(γ1/2) + tanh(γ1/2) = 0, m = 2, 4, 6,...For odd harmonics: X(x) = sin γ2(xa−12) −sin(γ2/2)sinh(γ2/2)sinhγ2(xa−12)tan(γ2/2) − tanh(γ2/2) = 0, m = 3, 5, 7,...(18.7)§18.2.3 Free at x = 0 and x = a (F-F):For m=0: X(x) = 1For m=1: X(x) = 1 −2xaFor even harmonics: X(x) = cos γ1(xa−12) −sin(γ1/2)sinh(γ1/2)coshγ1(xa−12)tan(γ1/2) + tanh(γ1/2) = 0, m = 2, 4, 6,...For odd harmonics: X(x) = sin γ1(xa−12) +sin(γ2/2)sinh(γ2/2)sinhγ2(xa−12)tan(γ2/2) − tanh(γ2/2) = 0, m = 3, 5, 7,...(18.8)The above assumed deflection satisfies only approximately the shear condition, i.e., the zero shear forcealong the free edges.§18.2.4 Clamped at x = 0 and Free at x = a (C-F):X(x) = (cosγ3xa− coshγ3xa) + (sinγ3− sinh γ3cosγ3− cosh γ3)(sinγ3xa− sinhγ3xa)cosγ3coshγ3+ 1 = 0, m = 1, 2, 3,...(18.9)§18.2.5 Clamped at x = 0 and Simply Supported at x = a (C - SS):X(x) = sin γ2(x2a−12) −sin(γ2/2)sinh(γ2/2)sinhγ2(x2a−12)tan(γ2/2) − tanh(γ2/2) = 0, m = 2, 3, 4,...(18.10)§18.2.6 Free at x = 0 and Simply Supported at x = a (F-SS):For m=1: X(x) = 1 −xaX(x) = sin γ2(x2a−12) +sin(γ2/2)sinh(γ2/2)sinhγ2(x2a−12)tan(γ2/2) − tanh(γ2/2) = 0, m = 2, 3, 4,...(18.11)Once again (m, n) denote the number of vibration nodal lines lying in the x and y-directions including theboundaries as thenodal lines except when theboundary is free. We now list some of the frequency parametersλmnfrom Leissa [1] for various boundary conditions for a square plate.18–3Chapter 18: CLASSICAL AND FEM SOLUTIONS OF PLATE VIBRATION18–4Table 1 Frequency Parameter λmn= ωmna2√ρ/D for SS-C-SS-C Square Plateλ11λ21λ12λ22λ31λ1328.946 54.743 69.320 94.584 102.213 129.086Table 2 Frequency Parameter λmn= ωmna2√ρ/D for SS-C-SS-SS Square Plateλ11λ21λ12λ22λ31λ1323.646 51.674 58.641 86.126 100.259 113.217Table 3 Frequency Parameter λmn= ωmna2√ρ/D for SS-C-SS-F Square Plate (ν = 0.3)λ11λ21λ12λ22λ31λ1312.69 33.06 41.70 63.01 72.40 90.61Table 4 Frequency Parameter λmn= ωmna2√ρ/D for SS-SS-SS-F Square Plate (ν = 0.3)λ11λ21λ12λ22λ31λ1311.68 27.76 41.20 59.07 61.86 90.2918–418–5 §18.2 ASSUMED VIBRATION MODE SHAPES FOR SIX BOUNDARY CONDITIONSTable 5 Frequency Parameter λmn= ωmna2√ρ/D for SS-F-SS-F Square Plate (ν = 0.3)λ11λ12λ13λ21λ22λ239.8696 16.13 36.72 39.48 46.74 70.75Table 6 Frequency Parameter λmn= ωmna2√ρ/D for C-C-C-C Square Plate (ν = 0.3)λ11λ21λ22λ31λ32λ4135.10 72.90 107.47 131.63 164.39 210.35Table 7 Frequency Parameter λmn= ωmna2√ρ/D for F-F-F-F Square Plate (ν = 0.3)First Mode Second Mode Third Mode Fourth Mode Fifth Mode Sixth Mode13.4728 19.5961 24.2702 35.1565 63.6870 77.5896Frequency parameters for other boundary conditions can be found in Leissa [1].18–5Chapter 18: CLASSICAL AND FEM SOLUTIONS OF PLATE VIBRATION18–6§18.3 FINITE ELEMENT ANALYSIS OF PLATE VIBRATIONBefore we utilize a typical finite element software for plate vibration analysis, it is instructive to understandhow the boundary conditions are treated. Let us begin with the simplest case, that is, the geometric boundarycondition. For plate, it is the clamped edge:W(x, y) = 0 and∂W(x, y)∂n= 0 (18.12)where n is the normal directional component from the clamped boundary edge.xy1234wθθxyabFig. 18.1 Plate bending element and degrees of freedom per nodeReferring to Figure 18.1, there are three discrete degrees of freedom per node, (w, θx,θy). Since we have,along the edges (x = 0, x = a),∂W(x,y)∂n=∂W(x,y)∂x= θy, the clamped edge boundary condition is satisfiedif one choosesalong (x = 0, x = a) : w = 0 and θy= 0along (y = 0, y = b) : w = 0 and θx= 0(18.13)However, simply supported boundary conditions must satisfyW(x, y) = 0 and∂2W(x, y)∂n2= 0 (18.14)This boundary conditions are approximately satisfied viaalong (x = 0, x = a) : w = 0 and θyis free to rotate.along (y = 0, y = b) : w = 0 and θxis free to rotate.(18.15)It is important to recognize thatalong (x = 0, x = a) :∂2W(x, y)∂n2=∂θy∂xalong (y = 0, y = b) :∂2W(x, y)∂n2=∂θx∂y(18.16)18–618–7 §18.3 FINITE ELEMENT ANALYSIS OF PLATE VIBRATIONHence, the second of the simply supported boundary conditions are approximately satisfied as the elementsize (a, b) becomes smaller and smaller. This can be seen from the finite difference expression of (18.16):along (x = 0, x = a) :∂2W(x, y)∂n2=∂θy∂x≈(θ(2)y+ θ(3)y)2a−(θ(1)y+ θ(4)y)2aalong (y = 0, y = b) :∂2W(x, y)∂n2=∂θx∂y≈(θ(3)x+ θ(4)x)2b−(θ(1)x+ θ(2)x)2b(18.17)where the superscript designates the element node as shown in Figure 18.1.Similarly, the