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Comments on “natural frequencies of square membrane and square plate with rounded corners”
Joumal of Sound and Vibration (1983) 91(4), 595-596
LETTERS TO THE EDITOR COMMENTS
ON “NATURAL FREQUENCIES OF SQUARE MEMBRANE SQUARE PLATE WITH ROUNDED CORNERS”
AND
The writers compliment the authors for their very accurate results for a type of geometric configuration which is important from a mechanical engineering view-point [l]. The authors have made use of the conformal mapping approach and the Ritz method to determine eigenvalues for several modes of vibration. It is the purpose of this letter to remind the interested reader that, if the conformal mapping method is used, there are alternative ways to calculate the eigenvalues which may prove useful when studying the lower modes of vibration. Admittedly the approach developed in reference [l] is more accurate and possesses considerably more capability of dynamic description of the behavior of the system. Determination of Sz2go’s upper bound in the case of a vibrating membrane. Making use of SzZgo’s bounding technique [2] one has (following the notation used in reference [ 11)
T
Jpl
mOla
s PO~IL,
(1)
where poI is the first root of the Bessel function of the first kind and zero order (pal = 2.4048) for the first SS mode. Making L = 25/24 and substituting in equation (1) one obtains dp/ T wO,a s 2.3086,
(2)
while the result obtained in reference [l] is A = 2.308. In the case of a square membrane JPIT
wOla G 2*4048/1*079 = 2.228,
(3)
which is a close upper bound for the exact eigenvalue. The reader can easily verify that a similar r-on is valid for higher modes. For instance, for the first antisymmetric mode Wlla c 3*8317/L = 3.678, which is in excellent agreement with the result SA-&p/T obtained in reference [l]. Use of polynomial co-ordinate functions. The governing differential equation in the l-plane is (S = re’“)
a2W 1 aw q+;
;+l
1 r
a2w
-+ ae2
(4)
Using the Galerkin method one makes W= Wa= f: bi(l-r2’). i=l
Substituting then expression (5) in equation (4) and making use of Galerkin’s minimization procedure one easily obtains, for I = 2, dp/ T q,,a
= 2.308.
595 0022-460X/83/240595+03
$03.00/O
@ 1983 Academic Press Inc. (London) Limited
596
LETTERS To THE tDIlWR
The procedure is quite similar in the case of clamped plates. For simply supported and free plates one must use the Ritz method since the natural boundary conditions are not satisfied. Institute of Applied Mechanics, Puerto Belgrano Naval Base, 8111 -Argentina
P. A. A. LAURA
Mechanical Systems Group, Facultad Regional Bahia Blanca (CJTN), Bahia Blanca, 8000-Argentina (Received
L. BACHA
DE NA~ALINI
7 May 1983) REFERENCES
1. T. IRIE, G. YAMADA and M. SONODA 1983 Journal of Sound and Vibration 86, 442-448. Natural frequencies of square membrane and square plate with rounded corners. 2. P. A. A. LAURA 1964 Journal of the Royal Aeronautical Society 67,274-275. On the determination of the natural frequency of a star shaped membrane.
LETTERS TO THE EDITOR COMMENTS
ON “NATURAL FREQUENCIES OF SQUARE MEMBRANE SQUARE PLATE WITH ROUNDED CORNERS”
AND
The writers compliment the authors for their very accurate results for a type of geometric configuration which is important from a mechanical engineering view-point [l]. The authors have made use of the conformal mapping approach and the Ritz method to determine eigenvalues for several modes of vibration. It is the purpose of this letter to remind the interested reader that, if the conformal mapping method is used, there are alternative ways to calculate the eigenvalues which may prove useful when studying the lower modes of vibration. Admittedly the approach developed in reference [l] is more accurate and possesses considerably more capability of dynamic description of the behavior of the system. Determination of Sz2go’s upper bound in the case of a vibrating membrane. Making use of SzZgo’s bounding technique [2] one has (following the notation used in reference [ 11)
T
Jpl
mOla
s PO~IL,
(1)
where poI is the first root of the Bessel function of the first kind and zero order (pal = 2.4048) for the first SS mode. Making L = 25/24 and substituting in equation (1) one obtains dp/ T wO,a s 2.3086,
(2)
while the result obtained in reference [l] is A = 2.308. In the case of a square membrane JPIT
wOla G 2*4048/1*079 = 2.228,
(3)
which is a close upper bound for the exact eigenvalue. The reader can easily verify that a similar r-on is valid for higher modes. For instance, for the first antisymmetric mode Wlla c 3*8317/L = 3.678, which is in excellent agreement with the result SA-&p/T obtained in reference [l]. Use of polynomial co-ordinate functions. The governing differential equation in the l-plane is (S = re’“)
a2W 1 aw q+;
;+l
1 r
a2w
-+ ae2
(4)
Using the Galerkin method one makes W= Wa= f: bi(l-r2’). i=l
Substituting then expression (5) in equation (4) and making use of Galerkin’s minimization procedure one easily obtains, for I = 2, dp/ T q,,a
= 2.308.
595 0022-460X/83/240595+03
$03.00/O
@ 1983 Academic Press Inc. (London) Limited
596
LETTERS To THE tDIlWR
The procedure is quite similar in the case of clamped plates. For simply supported and free plates one must use the Ritz method since the natural boundary conditions are not satisfied. Institute of Applied Mechanics, Puerto Belgrano Naval Base, 8111 -Argentina
P. A. A. LAURA
Mechanical Systems Group, Facultad Regional Bahia Blanca (CJTN), Bahia Blanca, 8000-Argentina (Received
L. BACHA
DE NA~ALINI
7 May 1983) REFERENCES
1. T. IRIE, G. YAMADA and M. SONODA 1983 Journal of Sound and Vibration 86, 442-448. Natural frequencies of square membrane and square plate with rounded corners. 2. P. A. A. LAURA 1964 Journal of the Royal Aeronautical Society 67,274-275. On the determination of the natural frequency of a star shaped membrane.