- Email: [email protected]
NAILING DOWN A VIBRATING MEMBRANE
Journal of Sound and
NAILING DOWN A VIBRATING MEMBRANE C. Y. WANG Department of Mathematics, Michigan State ;niversity, East ¸ansing, MI 48824, ;.S.A. (Received 26 February 2001)
In a previous letter [1], the exact solution of a vibrating annular membrane is considered. The asymptotic form for the fundamental frequency k can be shown to be n> (k ) 1 #O("ln c"\), (1) k"k # 2J (k ) "ln c" where the outer radius is one, c is the core radius, and k is the fundamental frequency of the unit circle without the constraint. When cP0 or when the inner core shrinks to zero (a centered pin-point constraint), the frequency is the same as that of a circular membrane without the constraint. Laura et al. [2, 3] showed that the higher frequencies of a circular membrane are also una!ected by a centered pin-point constraint. The problem was further discussed by Gottlieb [4], who noted that Rayleigh [5] had conjectured that this phenomena would apply to any number of pin-points on a membrane of any shape. The second term of equation (1) is the "rst correction due to the small, "nite size of the point constraint. Since (dk/dc)PR as cP0, the rise of frequency is singular as c is increased from zero. From reference [1] one can show that the increase from the unconstrained case is about 5% when c is 10\ and 10% when c is 10\. Since no real constraint can have in"nitesimal size, this correction term is important. The purpose of this letter is to "nd the size correction term in the case of an arbitrary vibration mode of an arbitrary membrane with an arbitrary number, locations, and (small) sizes of internal point constraints. The method used is similar to that of Rayleigh [5], and may be well known to some researchers. Consider a membrane of arbitrary shape with N small internal circular constraints with radius c ;1 centered at P . Let w and k solve the membrane problem without the G G constraints, and w and k solve the problem the constraints. Thus
w #kw "0,
w#kw"0. (2, 3) Multiply equation (2) by w and equation (3) by w . The di!erence is then integrated over the multiply-connected area of the membrane:
(w w !w w) dA#(k!k)
ww dA"0.
(4)
Using Green's second theorem and noting that w is zero on the outer boundary, w is zero both on the outer boundary and the internal constraints, equation (4) gives (k!k ) 0022-460X/01/440738#03 $35.00/0
, ww dA"!
*w w ds , *n G
(5) 2001 Academic Press
LETTERS TO THE EDITOR
739
where s is the boundary of the ith constraint and n is the outward normal of the domain. G Let (o , ) be polar co-ordinates centered at P . Then, on each constraint ds "c d , G G G G G G */*n"!*/*p , w +w " . Since c is small, w approximately satis"es the Laplace equation. G .G G The non-periodic solution is
ln o G , w+w " 1! .G ln c G
(6)
where the constants are adjusted such that w is zero on o "c , and w+w " when c P0. G G .G G The derivative on the boundary is thus *w !w " .G . " c ln c *o G G G
(7)
2n , (w " ) k"k# .G #2, w dA (!ln c ) G
(8)
Equation (5) then gives
where w+w in the area integral. The corresponding eigenfunction is ln o , G#2. (9) w"w ! w " .G ln c G For the fundamental mode of a circular boundary, w "J (k r) and equation (8) simpli"es to 2 J (k r ) k"k# G #2. J (k ) (!ln c ) G
(10)
For a single constraint at the center, r "0, c "c and equation (10) becomes G G 1 k"k # #2. k J (k )(!ln c)
(11)
which is identical to equation (1) after using the Wronskin identity J (z)> (z)!J (z)> (z)"2/nz.
(12)
We shall give some examples. Consider a square membrane of side length one and nailed at the center. For the fundamental mode k "(2n and w "sin(nx) sin(ny). From equation (8), the frequency with the constraint is k"2n#8n
1 #2, "ln c"
(13)
where c 1 is the radius of the constraint. Consider next a circular membrane nailed at seven evenly points: r "0, r "0)5, h "a#(i!2)n/3, c "c for i"2}7, Then for the G G G fundamental mode equation (10) gives 2 1 27)40 k"k# [1#6J(k /2)] #2"5.783# #2 J (k ) "ln c" "ln c"
(14)
740
LETTERS TO THE EDITOR
Now for the second mode k "3)8317 and w "sin h J (k r). Equation (8) gives 1 4 J (k /2) sin[a#(i!2)n/3] #2 k"k# J (k) "ln c" 24)95 "14)682# #2 "ln c"
(15)
For both modes, the phase angle a does not enter the frequency correction. Finally, guided by comparisons with the exact solution in reference [1], the constraint radius c should be less than about 0)001 for equation (8) to have an error or less than 1%. Also, the constraint locations should be more than O(c) distance from the membrane boundary and from each other.
REFERENCES 1. C. Y. WANG 1998 Journal of Sound and
NAILING DOWN A VIBRATING MEMBRANE C. Y. WANG Department of Mathematics, Michigan State ;niversity, East ¸ansing, MI 48824, ;.S.A. (Received 26 February 2001)
In a previous letter [1], the exact solution of a vibrating annular membrane is considered. The asymptotic form for the fundamental frequency k can be shown to be n> (k ) 1 #O("ln c"\), (1) k"k # 2J (k ) "ln c" where the outer radius is one, c is the core radius, and k is the fundamental frequency of the unit circle without the constraint. When cP0 or when the inner core shrinks to zero (a centered pin-point constraint), the frequency is the same as that of a circular membrane without the constraint. Laura et al. [2, 3] showed that the higher frequencies of a circular membrane are also una!ected by a centered pin-point constraint. The problem was further discussed by Gottlieb [4], who noted that Rayleigh [5] had conjectured that this phenomena would apply to any number of pin-points on a membrane of any shape. The second term of equation (1) is the "rst correction due to the small, "nite size of the point constraint. Since (dk/dc)PR as cP0, the rise of frequency is singular as c is increased from zero. From reference [1] one can show that the increase from the unconstrained case is about 5% when c is 10\ and 10% when c is 10\. Since no real constraint can have in"nitesimal size, this correction term is important. The purpose of this letter is to "nd the size correction term in the case of an arbitrary vibration mode of an arbitrary membrane with an arbitrary number, locations, and (small) sizes of internal point constraints. The method used is similar to that of Rayleigh [5], and may be well known to some researchers. Consider a membrane of arbitrary shape with N small internal circular constraints with radius c ;1 centered at P . Let w and k solve the membrane problem without the G G constraints, and w and k solve the problem the constraints. Thus
w #kw "0,
w#kw"0. (2, 3) Multiply equation (2) by w and equation (3) by w . The di!erence is then integrated over the multiply-connected area of the membrane:
(w w !w w) dA#(k!k)
ww dA"0.
(4)
Using Green's second theorem and noting that w is zero on the outer boundary, w is zero both on the outer boundary and the internal constraints, equation (4) gives (k!k ) 0022-460X/01/440738#03 $35.00/0
, ww dA"!
*w w ds , *n G
(5) 2001 Academic Press
LETTERS TO THE EDITOR
739
where s is the boundary of the ith constraint and n is the outward normal of the domain. G Let (o , ) be polar co-ordinates centered at P . Then, on each constraint ds "c d , G G G G G G */*n"!*/*p , w +w " . Since c is small, w approximately satis"es the Laplace equation. G .G G The non-periodic solution is
ln o G , w+w " 1! .G ln c G
(6)
where the constants are adjusted such that w is zero on o "c , and w+w " when c P0. G G .G G The derivative on the boundary is thus *w !w " .G . " c ln c *o G G G
(7)
2n , (w " ) k"k# .G #2, w dA (!ln c ) G
(8)
Equation (5) then gives
where w+w in the area integral. The corresponding eigenfunction is ln o , G#2. (9) w"w ! w " .G ln c G For the fundamental mode of a circular boundary, w "J (k r) and equation (8) simpli"es to 2 J (k r ) k"k# G #2. J (k ) (!ln c ) G
(10)
For a single constraint at the center, r "0, c "c and equation (10) becomes G G 1 k"k # #2. k J (k )(!ln c)
(11)
which is identical to equation (1) after using the Wronskin identity J (z)> (z)!J (z)> (z)"2/nz.
(12)
We shall give some examples. Consider a square membrane of side length one and nailed at the center. For the fundamental mode k "(2n and w "sin(nx) sin(ny). From equation (8), the frequency with the constraint is k"2n#8n
1 #2, "ln c"
(13)
where c 1 is the radius of the constraint. Consider next a circular membrane nailed at seven evenly points: r "0, r "0)5, h "a#(i!2)n/3, c "c for i"2}7, Then for the G G G fundamental mode equation (10) gives 2 1 27)40 k"k# [1#6J(k /2)] #2"5.783# #2 J (k ) "ln c" "ln c"
(14)
740
LETTERS TO THE EDITOR
Now for the second mode k "3)8317 and w "sin h J (k r). Equation (8) gives 1 4 J (k /2) sin[a#(i!2)n/3] #2 k"k# J (k) "ln c" 24)95 "14)682# #2 "ln c"
(15)
For both modes, the phase angle a does not enter the frequency correction. Finally, guided by comparisons with the exact solution in reference [1], the constraint radius c should be less than about 0)001 for equation (8) to have an error or less than 1%. Also, the constraint locations should be more than O(c) distance from the membrane boundary and from each other.
REFERENCES 1. C. Y. WANG 1998 Journal of Sound and