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Vibration analysis of a completely free elliptical plate
Journal of Sound and Vibration (I 974) 34(3), 441-443
LETTERS TO THE EDITOR VIBRATION
ANALYSIS
OF A COMPLETELY
FREE
ELLIPTICAL
PLATE
Although the free vibration of thin plates is considered a well known subject, there remain certain geometries which defy exact solution of the governing differential equation, or at least present formidable difficulty. The ellipse is one such geometry. Solutions to the differential equation take the form of an infinite series of both regular and modified Mathieu functions, which must be made to satisfy all the boundary conditions. Only two cases for the elliptical plate have been treated in the literature: the completely clamped and the simply supported. The completely clamped studies are presented in reference [1] and a solution for the simply supported in reference [2]. In both cases, the deflection function for the approximate methods was limited to a very small number of terms. Also, the "exact" solution for the completely clamped plate[l] was truncated after four terms of the infinite series. Thus, while the fundamental frequencies from these analyses are fairly accurate, one would expect higher frequencies and mode shapes to be in considerable error. The third homogeneous boundary condition is that of the completely free plate. For this case, the "exact" solution would be difficult. One would have to make this truncated series of regular and modified Mathieu functions satisfy the condition of zero shear and zero edge moment around the periphery. No previous analytical results have been found for this case, although the experimental work of Waller [3] was published in 1950. However, one can obtain as accurate a solution as one pleases by the Ritz method. For the assumed solution, choose a double power series, N --1 M -1
)v= 7_ i=o
7. X . r
(l)
J~O
where r = x/a, tl =y/b, a is the length of the semi-major axis and b is the length of the s~miminor axis. Equation (I) does not satisfy any boundary conditions since there are no ge.,metric constraints for a completely free plate. The shear and moment conditions will be satisfied in the limit [4] as more and more terms are taken in equation (I). When D is taken to be a constant, the following integral equation for an elliptical plate with a co-ordinate system at the center (see Figure I) results: D aZb 2
b2 a2 - - )V~6l~q~. + IV,,6W,, + v I V , , f l V ~
F
J
-1 -4i-s-~
t.12
-b2
t~ aZbZ
+ vIV~6Hq. + 2(I - v) W~,~6Wo) ~
phW61V
Y
Figure 1. Co-ordinate system for an elliptical plate. 441
]
dqd~=O,
(2)
442
LETTERS TO THE EDITOR
where the subscripts denote differentiation, D = Eh3/12(l - v2), h is the plate thickness, v is Poisson's ratio, p is the mass density, oJ the natural circular frequency, and ~5IV= (a tV/aAk,)• ~5//kz. 6Akt is arbitrary and may be factored out of equation (2). One then obtains N • M simultaneous algebraic equations in the N x M unknown ~4~-'s and to, with the standard eigenvalue problem resulting. Since double symmetry exists for the co-ordinate system chosen, four classes of uncoupled solutions result: (a) (b) (c) (d)
symmetric in 4, symmetric in tl; symmetric in ~, anti-symmetric in tl; anti-symmetric in 4, symmetric in ll; anti-symmetric in ~, anti-symmetric in t/.
For the results presented herein five i values and fivej values were used, yielding 25 terms in the double series for each of the four classes of solution. Tile integration of equation (2) was exact in the r/direction and numerical in the ~ direction. Gaussian quadrature was used for the numerical integration. The experimental work of Waller [3] is for a brass plate with a = 2.495 in, b = 1.26 in, h = 0.0638 in, v = 1/3, and thus a[b -- 1.98. The fundamental frequency was found by experiment to be 438 Hz [3]. However, E, Young's modulus, was not given in reference [3]. From the current analysis, the lowest eigenvalue was found to be 2 = 1.679, where ) z = to2phb4/D. The Bridgeport Brass Technical Handbook gives for high brass, E = 15 • 10 6 and p = 0-306 lb/in 3. Using these values in the calculated eigenvalue yields a calculated fundamental frequency of 452 Hz Which is well within the accuracy of the assumptions of bending theory. A comparison of frequency ratios is given in Table 1. The s and n notation refers to the apparent number of nodal lines in the ~ and tl directions, respectively. As mentioned earlier 25 terms were used for each symmetry class, yielding I00 modes and frequencies. Of these TABLE 1
Comparison of frequency ratios, a/b = 1-98; v = 1/3 Frequency ratio for an n value of A
s 0 1 2 3 4
0
1
0t [0]~t 0 [0] 0 [0] 1-55 [1.77] 4.14 [4.25] 6.41 [6-57] 10-5 [10.6] 13.8 [14.0] 19-9 [17.01 24.2 [22.0]
2
3
4
5
1.0 [1.0] 2.54 [2.58] 4.73 [4.71 7.56 [7.3] 3.26 [3.27] 5.47 [5:68] 8.25 [8.29] 11.6 [11.0] 9.24 [9.43] 12-7 [12.6]
6 1.6 [1o-o1
t Analytical results. :1:Experimental results of Waller [3] are in brackets.
100, at least the first ten for each symmetry class are well converged in frequency and mode shape. Thus, 40 of the 100 modes and frequencies are accurate from an analytical standpoint. The only significant discrepancies between the experimental and analytical frequency ratios in Table 1 are the modes (s = 4, n = 0) and (s = 4; n = 1). The method of the experimental frequency determination and excitation may be questionable, since these frequencies are on the order of 7500-9500 Hz. To further illustrate the accuracy of the analytical solution, correlation between several experimental photographs of nodal patterns and plotted analytical nodal patterns is presented in Plate I and Figure 2, respectively. The agreement is evident except for four modes.
.
-
aO
"2"
.
.
.
.
.
.
3
.
.
7
4
$
0 i
I
O O O
5
6
91P Plate 1. Experimental nodal patterns,
nO
l
2
a/b = 1.98.
5
4
5
0~~ 40~ Figure 2. Analytical n o d a l patterns, a / b = 1.98.
{facing p. 442)
LETYERSTO TIlE EDITOR
443
The first two discrepancies in mode shapes are for the modes (s = 2; n = 2) and (s = 2; n = 3). The experimental photographs of these two modes lack definition in the regions of intersection or close proximity of two nodal lines. In the corresponding analytical modes, these nodal lines are clearly defined. If these regions of close proximity were to coalesce, s would be 2 and n would be 2 and 3, respectively. Thus, these are called apparent s and n values. The only other differences are the modes (s = 4; n = 0) and (s = 4; n = 1). The experimental nodal patterns clearly lack definition in the tip regions of the plate, the reason possibly being the extremely high frequencies involved for these two modes, as discussed earlier. The corresponding analytical modes look the same toward the center but are also well defined near the tips of the plate. The analytical nodal patterns, for these fotir modes discussed, are well converged to the mathematical model of the system. Whether the assumptions of thin plate theory are breaking down for frequencies higher than 7500 Hz, or that the experimental techniques were inadequate at these high frequencies, is a matter for conjecture. From the experimental correlation presented, the accuracy of the Ritz method is quite apparent. The.author feels that these results should spur confidence in the accuracy of the Ritz method for solving thin plate vibration problems with difficult geometries. The results presented cost approximately $28 on the IBM 370 digital computer. ACKNOWLEDGMENT The author wishes to thank NASA Langley for their support of this work through research grant No. N G R 36-008-197 entitled "Application of the General Energy Equation--A Unified Approach to Mechanics", Dr C. D. Bailey, principal investigator. Aeronautical Enghzeerhtg Department, The Ohio State Unicersity, Cohtmbus, Ohio 43210, U.S.A. (Received 15 January 1974)
D. P. BERES
REFERENCES 1. A. W. LEISSA1969 NASA SP-160. Vibration of plates. 2. A. W. LElSSA 1967 Journal ofSotmd attd Vibration 6, 145-148. Vibration of a simply-supported elliptical plate. 3. M. D. WALLER1950 Proceedings of the Physical Society (London) Series B 63, 451---455.Vibration of free elliptical plates. 4. C. D. BAILEY1973 submitted to American Institute of Aeronautics attd Astronautics. Hamilton's law applied to stationary and non-stationary motion of beams.
LETTERS TO THE EDITOR VIBRATION
ANALYSIS
OF A COMPLETELY
FREE
ELLIPTICAL
PLATE
Although the free vibration of thin plates is considered a well known subject, there remain certain geometries which defy exact solution of the governing differential equation, or at least present formidable difficulty. The ellipse is one such geometry. Solutions to the differential equation take the form of an infinite series of both regular and modified Mathieu functions, which must be made to satisfy all the boundary conditions. Only two cases for the elliptical plate have been treated in the literature: the completely clamped and the simply supported. The completely clamped studies are presented in reference [1] and a solution for the simply supported in reference [2]. In both cases, the deflection function for the approximate methods was limited to a very small number of terms. Also, the "exact" solution for the completely clamped plate[l] was truncated after four terms of the infinite series. Thus, while the fundamental frequencies from these analyses are fairly accurate, one would expect higher frequencies and mode shapes to be in considerable error. The third homogeneous boundary condition is that of the completely free plate. For this case, the "exact" solution would be difficult. One would have to make this truncated series of regular and modified Mathieu functions satisfy the condition of zero shear and zero edge moment around the periphery. No previous analytical results have been found for this case, although the experimental work of Waller [3] was published in 1950. However, one can obtain as accurate a solution as one pleases by the Ritz method. For the assumed solution, choose a double power series, N --1 M -1
)v= 7_ i=o
7. X . r
(l)
J~O
where r = x/a, tl =y/b, a is the length of the semi-major axis and b is the length of the s~miminor axis. Equation (I) does not satisfy any boundary conditions since there are no ge.,metric constraints for a completely free plate. The shear and moment conditions will be satisfied in the limit [4] as more and more terms are taken in equation (I). When D is taken to be a constant, the following integral equation for an elliptical plate with a co-ordinate system at the center (see Figure I) results: D aZb 2
b2 a2 - - )V~6l~q~. + IV,,6W,, + v I V , , f l V ~
F
J
-1 -4i-s-~
t.12
-b2
t~ aZbZ
+ vIV~6Hq. + 2(I - v) W~,~6Wo) ~
phW61V
Y
Figure 1. Co-ordinate system for an elliptical plate. 441
]
dqd~=O,
(2)
442
LETTERS TO THE EDITOR
where the subscripts denote differentiation, D = Eh3/12(l - v2), h is the plate thickness, v is Poisson's ratio, p is the mass density, oJ the natural circular frequency, and ~5IV= (a tV/aAk,)• ~5//kz. 6Akt is arbitrary and may be factored out of equation (2). One then obtains N • M simultaneous algebraic equations in the N x M unknown ~4~-'s and to, with the standard eigenvalue problem resulting. Since double symmetry exists for the co-ordinate system chosen, four classes of uncoupled solutions result: (a) (b) (c) (d)
symmetric in 4, symmetric in tl; symmetric in ~, anti-symmetric in tl; anti-symmetric in 4, symmetric in ll; anti-symmetric in ~, anti-symmetric in t/.
For the results presented herein five i values and fivej values were used, yielding 25 terms in the double series for each of the four classes of solution. Tile integration of equation (2) was exact in the r/direction and numerical in the ~ direction. Gaussian quadrature was used for the numerical integration. The experimental work of Waller [3] is for a brass plate with a = 2.495 in, b = 1.26 in, h = 0.0638 in, v = 1/3, and thus a[b -- 1.98. The fundamental frequency was found by experiment to be 438 Hz [3]. However, E, Young's modulus, was not given in reference [3]. From the current analysis, the lowest eigenvalue was found to be 2 = 1.679, where ) z = to2phb4/D. The Bridgeport Brass Technical Handbook gives for high brass, E = 15 • 10 6 and p = 0-306 lb/in 3. Using these values in the calculated eigenvalue yields a calculated fundamental frequency of 452 Hz Which is well within the accuracy of the assumptions of bending theory. A comparison of frequency ratios is given in Table 1. The s and n notation refers to the apparent number of nodal lines in the ~ and tl directions, respectively. As mentioned earlier 25 terms were used for each symmetry class, yielding I00 modes and frequencies. Of these TABLE 1
Comparison of frequency ratios, a/b = 1-98; v = 1/3 Frequency ratio for an n value of A
s 0 1 2 3 4
0
1
0t [0]~t 0 [0] 0 [0] 1-55 [1.77] 4.14 [4.25] 6.41 [6-57] 10-5 [10.6] 13.8 [14.0] 19-9 [17.01 24.2 [22.0]
2
3
4
5
1.0 [1.0] 2.54 [2.58] 4.73 [4.71 7.56 [7.3] 3.26 [3.27] 5.47 [5:68] 8.25 [8.29] 11.6 [11.0] 9.24 [9.43] 12-7 [12.6]
6 1.6 [1o-o1
t Analytical results. :1:Experimental results of Waller [3] are in brackets.
100, at least the first ten for each symmetry class are well converged in frequency and mode shape. Thus, 40 of the 100 modes and frequencies are accurate from an analytical standpoint. The only significant discrepancies between the experimental and analytical frequency ratios in Table 1 are the modes (s = 4, n = 0) and (s = 4; n = 1). The method of the experimental frequency determination and excitation may be questionable, since these frequencies are on the order of 7500-9500 Hz. To further illustrate the accuracy of the analytical solution, correlation between several experimental photographs of nodal patterns and plotted analytical nodal patterns is presented in Plate I and Figure 2, respectively. The agreement is evident except for four modes.
.
-
aO
"2"
.
.
.
.
.
.
3
.
.
7
4
$
0 i
I
O O O
5
6
91P Plate 1. Experimental nodal patterns,
nO
l
2
a/b = 1.98.
5
4
5
0~~ 40~ Figure 2. Analytical n o d a l patterns, a / b = 1.98.
{facing p. 442)
LETYERSTO TIlE EDITOR
443
The first two discrepancies in mode shapes are for the modes (s = 2; n = 2) and (s = 2; n = 3). The experimental photographs of these two modes lack definition in the regions of intersection or close proximity of two nodal lines. In the corresponding analytical modes, these nodal lines are clearly defined. If these regions of close proximity were to coalesce, s would be 2 and n would be 2 and 3, respectively. Thus, these are called apparent s and n values. The only other differences are the modes (s = 4; n = 0) and (s = 4; n = 1). The experimental nodal patterns clearly lack definition in the tip regions of the plate, the reason possibly being the extremely high frequencies involved for these two modes, as discussed earlier. The corresponding analytical modes look the same toward the center but are also well defined near the tips of the plate. The analytical nodal patterns, for these fotir modes discussed, are well converged to the mathematical model of the system. Whether the assumptions of thin plate theory are breaking down for frequencies higher than 7500 Hz, or that the experimental techniques were inadequate at these high frequencies, is a matter for conjecture. From the experimental correlation presented, the accuracy of the Ritz method is quite apparent. The.author feels that these results should spur confidence in the accuracy of the Ritz method for solving thin plate vibration problems with difficult geometries. The results presented cost approximately $28 on the IBM 370 digital computer. ACKNOWLEDGMENT The author wishes to thank NASA Langley for their support of this work through research grant No. N G R 36-008-197 entitled "Application of the General Energy Equation--A Unified Approach to Mechanics", Dr C. D. Bailey, principal investigator. Aeronautical Enghzeerhtg Department, The Ohio State Unicersity, Cohtmbus, Ohio 43210, U.S.A. (Received 15 January 1974)
D. P. BERES
REFERENCES 1. A. W. LEISSA1969 NASA SP-160. Vibration of plates. 2. A. W. LElSSA 1967 Journal ofSotmd attd Vibration 6, 145-148. Vibration of a simply-supported elliptical plate. 3. M. D. WALLER1950 Proceedings of the Physical Society (London) Series B 63, 451---455.Vibration of free elliptical plates. 4. C. D. BAILEY1973 submitted to American Institute of Aeronautics attd Astronautics. Hamilton's law applied to stationary and non-stationary motion of beams.