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Comments on “free vibrations of circular arches: a review”
Journal of Sound and Vibration (1995) 187(1), 167–168
LETTERS TO THE EDITOR COMMENTS ON ‘‘FREE VIBRATIONS OF CIRCULAR ARCHES: A REVIEW’’ V. H. C´ Institute of Applied Mechanics (CONICET–SENID–ACCE) Gorriti 43, 8000 Bahı´ a Blanca, Argentina (Received 9 January 1995)
The authors have indeed presented a ‘‘critical review of the free dynamics of circular arches’’ [1]. They have succeeded in presenting interesting numerical information for the problem under consideration. Unfortunately, they have misunderstood some of the existing information in the available technical literature. 1. In their explanation of the Rayleigh–Schmidt method, they state that ‘‘the underlying philosophy is very simple: to the trial Rayleigh function another term must be added’’. They express then their equation (14) as w1 (a)=C1 a g+w(a). Apparently, this is not correct, because if w(a) is a trial Rayleigh function that satisfies the essential boundary conditions, w1 (a) does not necessarily comply with them. For instance, if w(a) satisfies the essential condition w(a0 )=0 (for a0$0), it is simple to see that w1 (a0 )=C1 a0$0. 2. The authors’ statement regarding the Galerkin method is not, certainly, the classical one. In fact, their equation (18) corresponds to the least squares method. In certain situations the Galerkin method is equivalent to the method of least squares, but this is not always the case. The least squares approach requires, in general, higher order continuity of the trial functions [2]. This is especially important when the trial functions have been selected in a piecewise way. It is not the goal of this discussion to represent a full treatment of the problem and the reader is referred to the classical literature. 3. The authors have misunderstood their reference [14] (reference [3] of this Letter). This reference deals with cantilever arches, as is clearly stated (even in the title), and the authors state that they compared results obtained in reference [3] of this Letter with the values corresponding to two hinged arches (their Table 1). On the other hand, column six of their Table 1 contains values which cannot be considered as ‘‘far lower results than the other ones’’. This is a matter of opinion, of course. In one case the result is even higher than other eigenvalues presented in Table 1, and for which an ‘‘exact’’ result is not available. It may be that the authors are referring to another paper by the writer and co-workers. The writer certainly humbly apologizes for any error which may have thus been incurred!
1. N. M. A and M. A. D R 1994 Journal of Sound and Vibration 176, 433–458. Free vibrations of circular arches: a review. 2. O. C. Z and K. M 1983 Finite Elements and Approximation. Singapore: John Wiley. See pp. 56 and 260. 3. V. H. C´ , P. A. A. L, C. P. F and R. C 1986 Journal of Sound and Vibration 110, 356–358. Numerical experiments on vibrating cantilever arches of varying cross-section. 167 0022–460X/95/410167+02 $12.00/0
7 1995 Academic Press Limited
LETTERS TO THE EDITOR COMMENTS ON ‘‘FREE VIBRATIONS OF CIRCULAR ARCHES: A REVIEW’’ V. H. C´ Institute of Applied Mechanics (CONICET–SENID–ACCE) Gorriti 43, 8000 Bahı´ a Blanca, Argentina (Received 9 January 1995)
The authors have indeed presented a ‘‘critical review of the free dynamics of circular arches’’ [1]. They have succeeded in presenting interesting numerical information for the problem under consideration. Unfortunately, they have misunderstood some of the existing information in the available technical literature. 1. In their explanation of the Rayleigh–Schmidt method, they state that ‘‘the underlying philosophy is very simple: to the trial Rayleigh function another term must be added’’. They express then their equation (14) as w1 (a)=C1 a g+w(a). Apparently, this is not correct, because if w(a) is a trial Rayleigh function that satisfies the essential boundary conditions, w1 (a) does not necessarily comply with them. For instance, if w(a) satisfies the essential condition w(a0 )=0 (for a0$0), it is simple to see that w1 (a0 )=C1 a0$0. 2. The authors’ statement regarding the Galerkin method is not, certainly, the classical one. In fact, their equation (18) corresponds to the least squares method. In certain situations the Galerkin method is equivalent to the method of least squares, but this is not always the case. The least squares approach requires, in general, higher order continuity of the trial functions [2]. This is especially important when the trial functions have been selected in a piecewise way. It is not the goal of this discussion to represent a full treatment of the problem and the reader is referred to the classical literature. 3. The authors have misunderstood their reference [14] (reference [3] of this Letter). This reference deals with cantilever arches, as is clearly stated (even in the title), and the authors state that they compared results obtained in reference [3] of this Letter with the values corresponding to two hinged arches (their Table 1). On the other hand, column six of their Table 1 contains values which cannot be considered as ‘‘far lower results than the other ones’’. This is a matter of opinion, of course. In one case the result is even higher than other eigenvalues presented in Table 1, and for which an ‘‘exact’’ result is not available. It may be that the authors are referring to another paper by the writer and co-workers. The writer certainly humbly apologizes for any error which may have thus been incurred!
1. N. M. A and M. A. D R 1994 Journal of Sound and Vibration 176, 433–458. Free vibrations of circular arches: a review. 2. O. C. Z and K. M 1983 Finite Elements and Approximation. Singapore: John Wiley. See pp. 56 and 260. 3. V. H. C´ , P. A. A. L, C. P. F and R. C 1986 Journal of Sound and Vibration 110, 356–358. Numerical experiments on vibrating cantilever arches of varying cross-section. 167 0022–460X/95/410167+02 $12.00/0
7 1995 Academic Press Limited