In the previous chapters, vibration behaviors of completely free and cantilevered rectangular plates with an edge V-notch have been determined via the present method. Some conclusions are drawn from the foregoing studies:
(1) The corner functions exactly satisfy the free boundary conditions along a V-notch and appropriately describe the stress singularity behaviors around the vertex of a V-notch. It has been demonstrated that the convergence of the numerical solutions can be accelerated by supplementing the admissible functions in the Ritz method with the corner functions.
(2) Matrix ill-conditioning occurs when the total number of admissible functions used is too large. Through the supplements of corner functions, one can obtain the convergent solutions with high accuracy (4 significant digit convergence) before the ill-conditioning occurs.
(3) It has been shown that poor convergence is obtained by using polynomial functions only when plates with a sharp V-notch. As the V-notch becomes deeper, it needs more number of corner functions to obtain accurate solutions.
(4) It has been shown that a shallow V-notch has only a small effect on the vibration behaviors of a V-notch plate. As the V-notch is deeper, frequencies significantly decrease mainly because of the reduction of the flexural stiffness, and the nodal patterns changes more violently. The
curve veering and the distortion of the straight nodal lines may occur due to the destruction of the symmetry when a V-notch exists. Sometimes, the modal order may exchange as the notch depth varies.
The thesis accurately determines vibration frequencies and nodal patterns of V-notched rectangular plates via the present method. These present results serve not only to improve the understanding the vibration behavior of a V-notched plate, but also as benchmark data against those from other numerical methods or experiments. The analysis methodology used here can be extended to other thin plate problems with stress singularities, such as a plate with a cut-out where more than one corners having stress singularities exist.
References
Aggarwala, B. D. and Ariel, P. D. (1981) “Vibration and bending of a cracked plate”, Rozprawy Inzynierskie, 29(2), pp. 295-310.
Filipich, C. P. and Rosales, M. B. (2000) “Arbitrary precision frequencies of a free rectangular thin plate”, Journal of Sound and Vibration, 230(3), pp. 521-539.
Hirano, Y. and Okazaki, K. (1980) “Vibration of cracked rectangular plates”, Bulletin of the Japan Society of Mechanical Engineers, 23(179), pp. 732-740.
Huang, C. S. (1991) Singularities in plate vibration problems, Ph. D.
dissertation, The Ohio State University, Columbus, Ohio.
Khadem, S. E. and Rezaee, M. (2000) “Introduction of modified comparison functions for vibration analysis of a rectangular cracked plate”, Journal of Sound and Vibration, 236(2), pp. 245-258.
Krawczuk, M (1993) “Natural vibrations of rectangular plates with a through crack”, Archive of Applied Mechanics, 63(7), pp. 491-504.
Leissa, A. W., McGee, O. G. and Huang, C. S. (1993) “Vibrations of Circular Plates Having V-notches or Sharp Radial Cracks”, Journal of
Sound and Vibration, 161(2), pp. 227-239.
Leissa, A. W., McGee, O. G. and Huang, C. S. (1993) “Vibrations of Sectorial Plates Having Corner Stress Singularities”, Journal of
Applied Mechanics, 60, pp. 136-140.
Leissa, A. W. (1973) “The free vibration of rectangular plates”, Journal of
Sound and Vibration, 31(3), pp. 257-293.
Leissa, A. W. (1969) Vibration of plates, NASA SP-160.
Liew, K. M., Hung, K. C. and Lim, M. K. (1994) “A solution method for analysis of cracked plates under vibration”, Engineering Fracture
Mechanics, 48(3), pp. 393-404.
Lynn, P. P. and Kumbasar, N. (1967) “Free vibrations of thin rectangular plates having narrow cracks with simply supported edges”,
Developments in Mechanics, 4, Proc. 10
thMidwestern Mechanics Conference, Colorado State University, Fort Collins, Colorado, August
21-23, pp. 911-928.Ma, C. C. and Huang, C. H. (2001) “Experimental and numerical analysis of vibrating cracked plates at resonant frequencies”, Experimental
Mechanics, 41(1), pp. 8-18.
Neku, K. (1982) “Free vibration of a simply-supported rectangular plate with a straight through-notch”, Bulletin of the Japan Society of
Mechanical Engineers, 25(199), pp. 16-23.
Qian, G. L., Gu, S. N. and Jiang, J. S. (1991) “A finite element model of cracked plates and application to vibration problems”, Computers and
Structures, 39(5), pp. 483-487.
Rossi, R. E. and Laura, P. A. A. (1996) “Symmetric and antisymmetric normal modes of a cantilever rectangular plate: effect of Poisson’s ratio and a concentrated mass”, Journal of Sound and Vibration, 195(1), pp.
142-148.
Solecki, R. (1983) “Bending vibration of a simply supported rectangular plate with a crack parallel to one edge”, Engineering Fracture
Mechanics, 18(6), pp. 1111-1118.
Stahl, B. and Keer, L. M. (1972) “Vibration and Stability of cracked rectangular plates”, International Journal of Solids and Structures, 8(1), pp. 69-91.
Timoshenko, S. and Woinowsky-Krieger, S. (1959), Theory of plates and
shells, 2
nd edition, McGraw-Hill.Williams, M. L. (1952) “Surface Stress Singularities Resulting from Various Boundary Conditions in Angular Corners of Plates under Bending”, Proceedings of the First U.S. National Congress of Applied
Mechanics, pp.325-329.
Yuan, J. and Dickinson, S. M. (1992) “The flexural vibration of rectangular plate systems approached by using artificial springs in the Rayleigh-Ritz method”, Journal of Sound and Vibration, 159(1), pp.
39-55
Table 3.1 Convergence of frequency parameters ωa2 ρh/D for a completely free square plate
order of polynomial (I×J)
Mode No.
3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 Leissa (1973)
Filipich and Rosales
(2000) 1 14.20 13.66 13.66 13.47 13.47 13.47 13.47 13.47 13.49 13.47 2 22.45 22.45 19.73 19.73 19.60 19.60 19.60 19.60 19.79 19.61 3 30.59 30.59 24.54 24.54 24.27 24.27 24.27 24.27 24.43 24.28 4 41.57 39.23 35.61 35.29 34.81 34.80 34.80 34.80 35.02 34.82 5 41.57 39.23 35.61 35.29 34.81 34.80 34.80 34.80 35.02 38.82
Table 3.2 Convergence of frequency parameters ωa2 ρh/D for a completely free square plate with a V-notch (c/a=0.5, d/b=0.03, α=170°)
order of polynomial (I×J) Mode
Table 3.3 Convergence of frequency parameters ωa2 ρh/D for a completely free square plate with a V-notch (c/a=0.5, d/b=0.1, α=30°)
order of polynomial (I×J) Mode
No.
No. of Corner
Functions 4×4 5×5 6×6 7×7 8×8 9×9 10×10
1
Note：“/”：no result due to matrix ill-conditioning
Table 3.4 Convergence of frequency parameters ωa2 ρh/D for a completely free square plate with a V-notch (c/a=0.5, d/b=0.3, α=30°)
order of polynomial (I×J) Mode
Table 3.5 Convergence of frequency parameters ωa2 ρh/D for a completely free square plate with a V-notch (c/a=0.5, d/b=0.5, α=30°)
order of polynomial (I×J) Mode
Table 3.6 Convergence of frequency parameters ωa2 ρh/D for a completely free square plate with a V-notch (c/a=0.5, d/b=0.3, α=5°)
order of polynomial (I×J) Mode
Table 3.7 Convergence of frequency parameters ωa2 ρh/D for a completely free square plate with a V-notch (c/a=0.5, d/b=0.3, α=0°)
order of polynomial (I×J) Mode
No.
No. of Corner
Functions 4×4 5×5 6×6 7×7 8×8 9×9 10×10
1
Table 3.8 Frequency parameters ωa2 ρh/D for completely free rectangular plates with a V-notch (a/b=1.0, υ=0.3)
D h a2 ρ / c/a α d/b ω
1 2 3 4 5 0* 13.47 19.60 24.27 34.80 34.80 0.1 13.31 19.40 24.08 34.20 34.21 0.3 11.58 17.63 22.91 28.31 32.09 5°
0.5 8.178 14.48 21.97 22.54 31.55 0.1 13.31 19.31 24.09 34.07 34.22 0.3 11.56 17.08 22.94 27.52 31.43 0.5
30°
0.5 7.843 13.82 21.21 21.88 30.92 0.1 13.36 19.52 24.20 34.19 34.64 0.3 12.07 18.38 22.80 27.61 33.75 5°
0.5 8.515 15.12 20.96 24.82 32.87 0.1 13.33 19.48 24.18 34.17 34.56 0.3 11.86 18.16 22.54 27.05 33.20 0.75
30°
0.5 7.961 14.60 20.57 24.37 32.27
Note：*：No V-notch
Table 3.9 Frequency parameters ωa2 ρh/D for completely free rectangular plates with a V-notch (a/b=2.0, υ=0.3)
D h a2 ρ / c/a α d/b ω
1 2 3 4 5 0* 21.46 26.57 58.48 59.61 88.01 0.1 21.29 26.36 58.19 59.46 87.97 0.3 19.96 24.47 56.63 57.00 87.60 5°
0.5 17.67 20.49 48.74 55.39 77.20 0.1 21.27 26.37 58.06 59.47 87.89 0.3 19.84 24.52 55.67 56.95 87.38 0.5
30°
0.5 17.38 20.37 47.51 53.73 74.19 0.1 21.40 26.44 58.09 59.17 87.96 0.3 20.83 25.23 53.45 56.03 87.57 5°
0.5 18.97 22.73 44.23 50.64 81.06 0.1 21.41 26.41 58.10 59.12 87.78 0.3 20.85 25.03 53.22 55.58 86.58 0.75
30°
0.5 18.95 22.28 42.54 49.54 77.86
Note：*：No V-notch
Table 3.10 Frequency parameters ωa2 ρh/D for completely free rectangular plates with a V-notch (a/b=0.5, υ=0.3)
D h a2 ρ / c/a α d/b ω
1 2 3 4 5 0* 5.366 6.644 14.62 14.90 22.00 0.1 5.356 6.561 14.13 14.86 21.02 0.3 4.963 5.316 10.07 13.92 15.07 5°
0.5 2.684 5.298 9.316 9.655 14.79 0.1 5.320 6.564 14.14 14.76 20.89 0.3 4.813 5.165 9.647 13.24 15.03 0.5
30°
0.5 2.428 5.135 9.081 9.366 14.61 0.1 5.357 6.584 14.28 14.88 21.56 0.3 5.213 5.330 9.226 14.60 15.22 5°
0.5 2.700 5.309 7.662 12.52 14.78 0.1 5.322 6.561 14.21 14.81 21.57 0.75
30°
0.3 4.864 5.247 8.712 14.19 15.20
Note：*：No V-notch
Table 3.11 Relative reductionsof the frequency parameters Δ ωn for completely free rectangular plates with a V-notch (a/b=1.0, υ=0.3)
ωn
Δ (%)
c/a α d/b
1 2 3 4 5 0.1 1.19 1.02 0.78 1.72 1.70
0.3 14.03 10.05 5.60 18.65 7.79 5°
0.5 39.29 26.12 9.48 35.23 9.34 0.1 1.19 1.48 0.74 2.10 1.67 0.3 14.18 12.86 5.48 20.92 9.68 0.5
30°
0.5 41.77 29.49 12.61 37.13 11.15 0.1 0.82 0.41 0.29 1.75 0.46 0.3 10.39 6.22 6.06 20.66 3.02 5°
0.5 36.79 22.86 13.64 28.68 5.55 0.1 1.04 0.61 0.37 1.81 0.69 0.3 11.95 7.35 7.13 22.27 4.60 0.75
30°
0.5 40.90 25.51 15.25 29.97 7.27
Table 3.12 Relative reductionsof the frequency parameters Δ ωn for completely free rectangular plates with a V-notch (a/b=2.0, υ=0.3)
ωn
Δ (%)
c/a α d/b
1 2 3 4 5 0.1 0.79 0.79 0.50 0.25 0.05
0.3 6.99 7.90 3.16 4.38 0.47 5°
0.5 17.66 22.88 16.66 7.08 12.28 0.1 0.89 0.75 0.72 0.23 0.14 0.3 7.55 7.72 4.81 4.46 0.72 0.5
30°
0.5 19.01 23.33 18.76 9.86 15.70 0.1 0.28 0.49 0.67 0.74 0.06 0.3 2.94 5.04 8.60 6.01 0.50 5°
0.5 11.60 14.45 24.37 15.05 7.90 0.1 0.23 0.60 0.65 0.82 0.26 0.3 2.84 5.80 8.99 6.76 1.62 0.75
30°
0.5 11.70 16.15 27.26 16.89 11.53
Table 3.13 Relative reductionsof the frequency parameters Δ ωn for completely free rectangular plates with a V-notch (a/b=0.5, υ=0.3)
ωn
Δ (%)
c/a α d/b
1 2 3 4 5 0.1 0.19 1.25 3.35 0.27 4.45
0.3 7.51 19.99 31.12 6.58 31.50 5°
0.5 49.98 20.26 36.28 35.20 32.77 0.1 0.86 1.20 3.28 0.94 5.05 0.3 10.31 22.26 34.02 11.14 31.68 0.5
30°
0.5 54.75 22.71 37.89 37.14 33.59 0.1 0.17 0.90 2.33 0.13 2.00 0.3 2.85 19.78 36.89 2.01 30.82 5°
0.5 49.68 20.09 47.59 15.97 32.82 0.1 0.82 1.25 2.80 0.60 1.95 0.75
30°
0.3 9.36 21.03 40.41 4.77 30.91
Table 4.1 Convergence of frequency parameters ωa2 ρh/D for a cantilevered square plate
order of polynomial (I×J)
Mode No.
3×3 4×4 5×5 6×6 7×7 8×8 9×9 10×10 Leissa (1973)
Rossi and Laura (1996) 1 3.494 3.489 3.475 3.474 3.472 3.472 3.471 3.471 3.492 3.471 2 8.597 8.546 8.544 8.513 8.512 8.509 8.509 8.508 8.525 8.508 3 21.56 21.50 21.31 21.31 21.29 21.29 21.29 21.29 21.43 21.29 4 31.41 31.32 27.46 27.46 27.20 27.20 27.20 27.20 27.33 27.20 5 32.42 31.33 31.20 30.98 30.98 30.97 30.96 30.96 31.11 30.96
Table 4.2 Convergence of frequency parameters ωa2 ρh/D for a cantilevered square plate with a V-notch (c/a=0.5, d/b=0.03, α=170°)
order of polynomial (I×J) Mode
Note：“/”：no result due to matrix ill-conditioning
Table 4.3 Convergence of frequency parameters ωa2 ρh/D for a cantilevered square plate with a V-notch (c/a=0.5, d/b=0.1, α=30°)
order of polynomial (I×J) Mode
No.
No. of Corner
Functions 4×4 5×5 6×6 7×7 8×8 9×9 10×10
1
Note：“/”：no result due to matrix ill-conditioning
Table 4.4 Convergence of frequency parameters ωa2 ρh/D for a cantilevered square plate with a V-notch (c/a=0.5, d/b=0.3, α=30°)
order of polynomial (I×J) Mode
No.
No. of Corner
Functions 4×4 5×5 6×6 7×7 8×8 9×9 10×10
1
Table 4.5 Convergence of frequency parameters ωa2 ρh/D for a cantilevered square plate with a V-notch (c/a=0.5, d/b=0.5, α=30°)
order of polynomial (I×J) Mode
Table 4.6 Convergence of frequency parameters ωa2 ρh/D for a cantilevered square plate with a V-notch (c/a=0.5, d/b=0.3, α=5°)
order of polynomial (I×J) Mode
No.
No. of Corner
Functions 4×4 5×5 6×6 7×7 8×8 9×9 10×10
1
Table 4.7 Convergence of frequency parameters ωa2 ρh/D for a cantilevered square plate with a straight crack (c/a=0.5, d/b=0.25, α=0°,
h/a=1/80, υ=0.33)
order of polynomial (I×J) ModeNote：“/”：no result due to matrix ill-conditioning
Table 4.8 Frequency parameters ωa2 ρh/D for cantilevered rectangular plates with a V-notch (a/b=1.0, υ=0.3)
D h a2 ρ / c/a α d/b ω
1 2 3 4 5 0* 3.471 8.508 21.29 27.20 30.96 0.1 3.459 8.407 20.96 27.01 30.47 0.3 3.348 7.511 18.66 24.96 28.83 5°
0.5 3.046 5.998 15.79 20.93 28.15 0.1 3.458 8.393 20.88 26.99 30.33 0.3 3.343 7.415 18.00 24.70 28.13 0.5
30°
0.5 3.026 5.799 14.81 19.16 27.21 0.1 3.469 8.449 21.16 27.06 30.62 0.3 3.436 7.923 19.28 23.57 28.01 5°
0.5 3.302 6.705 13.31 20.89 27.32 0.1 3.462 8.406 21.15 26.93 30.63 0.3 3.378 7.670 19.07 22.95 27.53 0.75
30°
0.5 3.132 6.332 12.30 20.49 26.64
Note：*：No V-notch
Table 4.9 Frequency parameters ωa2 ρh/D for cantilevered rectangular plates with a V-notch (a/b=2.0, υ=0.3)
D h a2 ρ / c/a α d/b ω
1 2 3 4 5 0* 3.440 14.80 21.44 48.19 60.16 0.1 3.434 14.71 21.29 47.94 60.05 0.3 3.384 13.94 20.20 46.10 57.86 5°
0.5 3.270 12.44 18.41 41.06 51.33 0.1 3.434 14.70 21.27 47.84 60.05 0.3 3.381 13.84 20.03 45.41 57.80 0.5
30°
0.5 3.263 12.22 17.99 39.32 50.17 0.1 3.439 14.76 21.38 47.82 59.71 0.3 3.427 14.42 20.86 43.90 56.19 5°
0.5 3.396 13.68 19.42 35.49 50.42 0.1 3.435 14.72 21.38 47.83 59.65 0.3 3.398 14.18 20.87 43.85 55.54 0.75
30°
0.5 3.318 13.24 19.37 34.46 49.05
Note：*：No V-notch
Table 4.10 Frequency parameters ωa2 ρh/D for cantilevered rectangular plates with a V-notch (a/b=0.5, υ=0.3)
D h a2 ρ / c/a α d/b ω
1 2 3 4 5 0* 3.493 5.352 10.18 19.08 21.84 0.1 3.468 5.262 9.897 18.59 21.02 0.3 3.068 4.220 8.379 15.10 17.04 5°
0.5 1.903 3.628 8.007 11.51 15.23 0.1 3.467 5.249 9.855 18.42 20.87 0.3 3.010 4.123 8.149 14.02 16.00 0.5
30°
0.5 1.750 3.598 7.828 10.91 13.33 0.1 3.487 5.299 10.01 18.87 21.41 0.3 3.302 4.467 8.396 11.74 18.38 5°
0.5 2.108 3.847 7.251 9.740 16.98 0.1 3.474 5.249 9.907 18.68 21.42 0.75
30°
0.3 3.097 4.264 8.023 10.84 18.04
Note：*：No V-notch
Table 4.11 Relative reductionsof the frequency parameters Δωn for cantilevered rectangular plates with a V-notch (a/b=1.0, υ=0.3)
ωn
Δ (%)
c/a α d/b
1 2 3 4 5 0.1 0.35 1.19 1.55 0.70 1.58
0.3 3.54 11.72 12.35 8.24 6.88 5°
0.5 12.24 29.50 25.83 23.05 9.08 0.1 0.37 1.35 1.93 0.77 2.03 0.3 3.69 12.85 15.45 9.19 9.14 0.5
30°
0.5 12.82 31.84 30.44 29.56 12.11 0.1 0.06 0.69 0.61 0.51 1.10 0.3 1.01 6.88 9.44 13.35 9.53 5°
0.5 4.87 21.19 37.48 23.20 11.76 0.1 0.26 1.20 0.66 0.99 1.07 0.3 2.68 9.85 10.43 15.63 11.08 0.75
30°
0.5 9.77 25.58 42.23 24.67 13.95
Table 4.12 Relative reductions Δ ωn of the frequency parameters for cantilevered rectangular plates with a V-notch (a/b=2.0, υ=0.3)
ωn
Δ (%)
c/a α d/b
1 2 3 4 5 0.1 0.17 0.61 0.70 0.52 0.18
0.3 1.63 5.81 5.78 4.34 3.82 5°
0.5 4.94 15.95 14.13 14.80 14.68 0.1 0.17 0.68 0.79 0.73 0.18 0.3 1.72 6.49 6.58 5.77 3.92 0.5
30°
0.5 5.15 17.43 16.09 18.41 16.61 0.1 0.03 0.27 0.28 0.77 0.75 0.3 0.38 2.57 2.71 8.90 6.60 5°
0.5 1.28 7.57 9.42 26.35 16.19 0.1 0.15 0.54 0.28 0.75 0.85 0.3 1.22 4.19 2.66 9.01 7.68 0.75
30°
0.5 3.55 10.54 9.65 28.49 18.47
Table 4.13 Relative reductions Δ ωn of the frequency parameters for cantilevered rectangular plates with a V-notch (a/b=0.5, υ=0.3)
ωn
Δ (%)
c/a α d/b
1 2 3 4 5 0.1 0.72 1.68 2.78 2.57 3.75
0.3 12.17 21.15 17.69 20.86 21.98 5°
0.5 45.52 32.21 21.35 39.68 30.27 0.1 0.74 1.92 3.19 3.46 4.44 0.3 13.83 22.96 19.95 26.52 26.74 0.5
30°
0.5 49.90 32.77 23.10 42.82 38.97 0.1 0.17 0.99 1.67 1.10 1.97 0.3 5.47 16.54 17.52 38.47 15.84 5°
0.5 39.65 28.12 28.77 48.95 22.25 0.1 0.54 1.92 2.68 2.10 1.92 0.75
30°
0.3 11.34 20.33 21.19 43.19 17.40
Fig. 2.1 Stress resultants in polar coordinate
Fig. 2.2 A sectorial plate
Fig. 2.3 The coordinate system defined in a sectorial plate
Fig. 2.4 Variation of minimum Re(λn
) with vertex angle α
(after Huang, C.S. (1991), Singularities in plate vibration problems, Ph. D dissertation, The Ohio State University, Columbus, Ohio.)
Fig. 3.1 The coordinate system defined in a completely free rectangular plate with a V-notch
Mode No.
α d/b
1 2 3 4 5
0
(13.47) (19.60) (24.27) (34.80) (34.80)
0.1
(13.31) (19.40) (24.08) (34.20) (34.21)
0.3
(11.58) (17.63) (22.91) (28.31) (32.09) 5°
0.5
(8.178) (14.48) (21.97) (22.54) (31.55)
0.1
(13.31) (19.31) (24.09) (34.07) (34.22)
0.3
(11.56) (17.08) (22.94) (27.52) (31.43) 30°
0.5
(7.843) (13.82) (21.21) (21.88) (30.92)
Fig. 3.2 Nodal patterns for completely free square plates with a V-notch at
c/a=0.5
Mode No.
α d/b
1 2 3 4 5
0
(13.47) (19.60) (24.27) (34.80) (34.80)
0.1
(13.36) (19.52) (24.20) (34.19) (34.64)
0.3
(12.07) (18.38) (22.80) (27.61) (33.75) 5°
0.5
(8.515) (15.12) (20.96) (24.82) (32.87)
0.1
(13.33) (19.48) (24.18) (34.17) (34.56)
0.3
(11.86) (18.16) (22.54) (27.05) (33.20) 30°
0.5
(7.961) (14.60) (20.57) (24.37) (32.27)
Fig. 3.3 Nodal patterns for completely free square plates with a V-notch at
c/a=0.75
Mode No.
α d/b
1 2 3 4 5
0
(21.46) (26.57) (58.48) (59.61) (88.01)
0.1
(21.29) (26.36) (58.19) (59.46) (87.97)
0.3
(19.96) (24.47) (56.63) (57.00) (87.60)
5°
0.5
(17.67) (20.49) (48.74) (55.39) (77.20)
Fig. 3.4 Nodal patterns for completely free rectangular plates (a/b=2.0) with a V-notch at c/a=0.5
Mode No.
α d/b
1 2 3 4 5
0
(21.46) (26.57) (58.48) (59.61) (88.01)
0.1
(21.27) (26.37) (58.06) (59.47) (87.89)
0.3
(19.84) (24.52) (55.67) (56.95) (87.38)
30°
0.5
(17.38) (20.37) (47.51) (53.73) (74.19)
Fig. 3.4 (continue)
Mode No.
α d/b
1 2 3 4 5
0
(21.46) (26.57) (58.48) (59.61) (88.01)
0.1
(21.40) (26.44) (58.09) (59.17) (87.96)
0.3
(20.83) (25.23) (53.45) (56.03) (87.57)
5°
0.5
(18.97) (22.73) (44.23) (50.64) (81.06)
Fig. 3.5 Nodal patterns for completely free rectangular plates (a/b=2.0) with a V-notch at c/a=0.75
Mode No.
α d/b
1 2 3 4 5
0
(21.46) (26.57) (58.48) (59.61) (88.01)
0.1
(21.41) (26.41) (58.10) (59.12) (87.78)
0.3
(20.85) (25.03) (53.22) (55.58) (86.58)
30°
0.5
(18.95) (22.28) (42.54) (49.54) (77.86)
Fig. 3.5 (continue)
Mode No.
α d/b
1 2 3 4 5
0
(5.366) (6.644) (14.42) (14.90) (22.00)
0.1
(5.356) (6.561) (14.13) (14.86) (21.02)
0.3
(4.963) (5.316) (10.07) (13.92) (15.07) 5°
0.5
(2.684) (5.298) (9.316) (9.655) (14.79)
Fig. 3.6 Nodal patterns for completely free rectangular plates (a/b=0.5) with a V-notch at c/a=0.5
Mode No.
α d/b
1 2 3 4 5
0
(5.366) (6.644) (14.42) (14.90) (22.00)
0.1
(5.320) (6.564) (14.14) (14.76) (20.89)
0.3
(4.813) (5.165) (9.647) (13.24) (15.03) 30°
0.5
(2.428) (5.135) (9.081) (9.366) (14.61)
Fig. 3.6 (continue)
Mode No.
α d/b
1 2 3 4 5
0
(5.366) (6.644) (14.42) (14.90) (22.00)
0.1
(5.357) (6.584) (14.28) (14.88) (21.56)
0.3
(5.213) (5.330) (9.226) (14.60) (15.22) 5°
0.5
(2.700) (5.309) (7.662) (12.52) (14.78)
Fig. 3.7 Nodal patterns for completely free rectangular plates (a/b=0.5) with a V-notch at c/a=0.75
Mode No.
α d/b
1 2 3 4 5
0
(5.366) (6.644) (14.42) (14.90) (22.00)
0.1
(5.322) (6.561) (14.21) (14.81) (21.57) 30°
0.3
(4.864) (5.247) (8.712) (14.19) (15.20)
Fig. 3.7 (continue)
Mode No.
k 4 5
0
0.38
-0.45
Note：“k=0”：the solutions obtained from 12×12 polynomials with the Ritz method.
Fig. 3.8 superposition of the fifth and forth mode shapes for completely free square plates
Fig. 4.1 The coordinate system defined in a cantilevered rectangular plate with a V-notch
Mode No.
α d/b
1 2 3 4 5
0
(3.471) (8.508) (21.29) (27.20) (30.96)
0.1
(3.459) (8.407) (20.96) (27.01) (30.47)
0.3
(3.348) (7.511) (18.66) (24.96) (28.83) 5°
0.5
(3.046) (5.998) (15.79) (20.93) (28.15)
0.1
(3.458) (8.393) (20.88) (26.99) (30.33)
0.3
(3.343) (7.415) (18.00) (24.70) (28.13) 30°
0.5
(3.026) (5.799) (14.81) (19.16) (27.21)
Fig. 4.2 Nodal patterns for cantilevered square plates with a V-notch at
c/a=0.5
Mode No.
α d/b
1 2 3 4 5
0
(3.471) (8.508) (21.29) (27.20) (30.96)
0.1
(3.469) (8.449) (21.16) (27.06) (30.62)
0.3
(3.436) (7.923) (19.28) (23.57) (28.01) 5°
0.5
(3.302) (6.705) (13.31) (20.89) (27.32)
0.1
(3.462) (8.406) (21.15) (26.93) (30.63)
0.3
(3.378) (7.670) (19.07) (22.95) (27.53) 30°
0.5
(3.132) (6.332) (12.30) (20.49) (26.64)
Fig. 4.3 Nodal patterns for cantilevered square plates with a V-notch at
c/a=0.75
Mode No.
α d/b
1 2 3 4 5
0
(3.440) (14.80) (21.44) (48.19) (60.16)
0.1
(3.434) (14.71) (21.29) (47.94) (60.05)
0.3
(3.384) (13.94) (20.20) (46.10) (57.86)
5°
0.5
(3.270) (12.44) (18.41) (41.06) (51.33)
Fig. 4.4 Nodal patterns for cantolevered rectangular plates (a/b=2.0) with a V-notch at c/a=0.5
Mode No.
α d/b
1 2 3 4 5
0
(3.440) (14.80) (21.44) (48.19) (60.16)
0.1
(3.434) (14.70) (21.27) (47.84) (60.05)
0.3
(3.381) (13.84) (20.03) (45.41) (57.80)
30°
0.5
(3.263) (12.22) (17.99) (39.32) (50.17)
Fig. 4.4 (continued)
Mode No.
α d/b
1 2 3 4 5
0
(3.440) (14.80) (21.44) (48.19) (60.16)
0.1
(3.439) (14.76) (21.38) (47.82) (59.71)
0.3
(3.427) (14.42) (20.86) (43.90) (56.19)
5°
0.5
(3.396) (13.68) (19.42) (35.49) (50.42)
Fig. 4.5 Nodal patterns for cantilevered rectangular plates (a/b=2.0) with a V-notch at c/a=0.75
Mode No.
α d/b
1 2 3 4 5
0
(3.440) (14.80) (21.44) (48.19) (60.16)
0.1
(3.435) (14.72) (21.38) (47.83) (59.65)
0.3
(3.398) (14.18) (20.87) (43.85) (55.54)
30°
0.5
(3.318) (13.24) (19.37) (34.46) (49.05)
Fig. 4.5 (continued)
Mode No.
α d/b
1 2 3 4 5
0
(3.493) (5.352) (10.18) (19.08) (21.84)
0.1
(3.468) (5.262) (9.897) (18.59) (21.02)
0.3
(3.068) (4.220) (8.379) (15.10) (17.04) 5°
0.5
(1.903) (3.628) (8.007) (11.51) (15.23)
Fig. 4.6 Nodal patterns for cantilevered rectangular plates (a/b=0.5) with a V-notch at c/a=0.5
Mode No.
α d/b
1 2 3 4 5
0
(3.493) (5.352) (10.18) (19.08) (21.84)
0.1
(3.467) (5.249) (9.855) (18.42) (20.87)
0.3
(3.010) (4.123) (8.149) (14.02) (16.00) 30°
0.5
(1.750) (3.598) (7.828) (10.91) (13.33)
Fig. 4.6 (continued)
Mode No.
α d/b
1 2 3 4 5
0
(5.366) (6.644) (14.42) (14.90) (22.00)
0.1
(3.487) (5.299) (10.01) (18.87) (21.41)
0.3
(3.302) (4.467) (8.396) (11.74) (18.38) 5°
0.5
(2.108) (3.847) (7.251) (9.740) (16.98)
Fig. 4.7 Nodal patterns for cantilevered rectangular plates (a/b=0.5) with a V-notch at c/a=0.75
Mode No.
α d/b
1 2 3 4 5
0
(3.493) (5.352) (10.18) (19.08) (21.84)
0.1
(3.474) (5.249) (9.907) (18.68) (21.42) 30°
0.3
(3.097) (4.264) (8.023) (10.84) (18.04)
Fig. 4.7 (continued)