Page 1 Vibrations of Structures Module IV: Vibrations of Membranes Lesson 32: The Rectangular Membrane Contents: 1. Modal Analysis 2. Modal Degeneracy Keywords: Rectangular membrane vibrations, Modal analysis, Degeneracy Page 2 Vibrations of Structures Module IV: Vibrations of Membranes Lesson 32: The Rectangular Membrane Contents: 1. Modal Analysis 2. Modal Degeneracy Keywords: Rectangular membrane vibrations, Modal analysis, Degeneracy The Rectangular Membrane 1 Modal Analysis x y (1; 0) T (1; 0) T (0; 1) T (0;1) T a b Figure 1: A rectangular membrane in cartesian coordinates Consider a rectangular membrane with all edges xed as shown in Fig. 1, and governed by w ;tt T(w ;xx + w ;yy ) = 0; (1) w(0; y; t) = 0; w(a; y; t) = 0; w(x; 0; t) = 0; w(x; b; t) = 0: (2) Assume a solution of the form w(x; y; t) = W(x; y)e i!t ; (3) where W(x; y) is the eigenfunction, and ! is the circular eigenfrequency. Substituting this solution into the equation of motion leads to the eigenvalue 2 Page 3 Vibrations of Structures Module IV: Vibrations of Membranes Lesson 32: The Rectangular Membrane Contents: 1. Modal Analysis 2. Modal Degeneracy Keywords: Rectangular membrane vibrations, Modal analysis, Degeneracy The Rectangular Membrane 1 Modal Analysis x y (1; 0) T (1; 0) T (0; 1) T (0;1) T a b Figure 1: A rectangular membrane in cartesian coordinates Consider a rectangular membrane with all edges xed as shown in Fig. 1, and governed by w ;tt T(w ;xx + w ;yy ) = 0; (1) w(0; y; t) = 0; w(a; y; t) = 0; w(x; 0; t) = 0; w(x; b; t) = 0: (2) Assume a solution of the form w(x; y; t) = W(x; y)e i!t ; (3) where W(x; y) is the eigenfunction, and ! is the circular eigenfrequency. Substituting this solution into the equation of motion leads to the eigenvalue 2 problem of the Helmholz equation r 2 W + ! 2 c 2 W = 0 (4) W(0; y) = 0; W(a; y) = 0; W(x; 0) = 0; and W(x; b) = 0 (5) whererW = W ;xx + W ;yy . Assume a solution of (4) of the form W(x; y) = Be i(k x x+k y y) ; (6) where B is a complex constant. Substituting (6) in (4) yields the dispersion relation of the membrane as k 2 x k 2 y + ! 2 c 2 = 0: (7) The solutions of (7) are of the form k x = and k y = such that 2 + 2 = ! 2 =c 2 . The general solution of (4) can then be written as W(x; y) = (B 1 e ix + B 2 e ix )(B 3 e iy + B 4 e iy ); (8) or in the form W(x; y) = A 1 cos x cosy + A 2 cos x siny + A 3 sin x cosy + A 4 sin x siny: (9) Using (9) in the rst and third boundary conditions in (5), one obtains A 1 cos y + A 2 sin y = 0 (10) and A 1 cos x + A 3 sin x = 0: (11) 3 Page 4 Vibrations of Structures Module IV: Vibrations of Membranes Lesson 32: The Rectangular Membrane Contents: 1. Modal Analysis 2. Modal Degeneracy Keywords: Rectangular membrane vibrations, Modal analysis, Degeneracy The Rectangular Membrane 1 Modal Analysis x y (1; 0) T (1; 0) T (0; 1) T (0;1) T a b Figure 1: A rectangular membrane in cartesian coordinates Consider a rectangular membrane with all edges xed as shown in Fig. 1, and governed by w ;tt T(w ;xx + w ;yy ) = 0; (1) w(0; y; t) = 0; w(a; y; t) = 0; w(x; 0; t) = 0; w(x; b; t) = 0: (2) Assume a solution of the form w(x; y; t) = W(x; y)e i!t ; (3) where W(x; y) is the eigenfunction, and ! is the circular eigenfrequency. Substituting this solution into the equation of motion leads to the eigenvalue 2 problem of the Helmholz equation r 2 W + ! 2 c 2 W = 0 (4) W(0; y) = 0; W(a; y) = 0; W(x; 0) = 0; and W(x; b) = 0 (5) whererW = W ;xx + W ;yy . Assume a solution of (4) of the form W(x; y) = Be i(k x x+k y y) ; (6) where B is a complex constant. Substituting (6) in (4) yields the dispersion relation of the membrane as k 2 x k 2 y + ! 2 c 2 = 0: (7) The solutions of (7) are of the form k x = and k y = such that 2 + 2 = ! 2 =c 2 . The general solution of (4) can then be written as W(x; y) = (B 1 e ix + B 2 e ix )(B 3 e iy + B 4 e iy ); (8) or in the form W(x; y) = A 1 cos x cosy + A 2 cos x siny + A 3 sin x cosy + A 4 sin x siny: (9) Using (9) in the rst and third boundary conditions in (5), one obtains A 1 cos y + A 2 sin y = 0 (10) and A 1 cos x + A 3 sin x = 0: (11) 3 This implies A 1 = 0, A 2 = 0 and A 3 = 0, and hence W(x; y) = A 4 sinx sin y: (12) The second and fourth boundary conditions in (5) now lead to sin a sin y = 0 and sinx sin b = 0 (13) which requires = m a ; and = n b ; m; n = 1; 2; : : :;1: (14) The eigenfunctions are then obtained from (12) as W (m;n) = sin mx a sin nx b ; m; n = 1; 2; : : :;1; (15) where W (m;n) represents the eigenfunction of the (m; n) mode. These eigen- functions satisfy the orthogonality condition hW (m;n) (x; y); W (r;s) (x; y)i = Z a 0 Z b 0 W (m;n) (x; y)W (r;s) (x; y) dx dy = ab 4 mr ns : (16) The rst few mode-shapes of the membrane are shown in Fig. 2. Using (14) in the condition 2 + 2 = ! 2 =c 2 yields the frequency equation ! (m;n) = c r m 2 a 2 + n 2 b 2 ; (17) where ! (m;n) represents the circular eigenfrequency of the (m; n) mode. Note that the eigenfrequencies of the membrane are not, in general, integral mul- tiples of the fundamental frequency (as in the case of a string). 4 Page 5 Vibrations of Structures Module IV: Vibrations of Membranes Lesson 32: The Rectangular Membrane Contents: 1. Modal Analysis 2. Modal Degeneracy Keywords: Rectangular membrane vibrations, Modal analysis, Degeneracy The Rectangular Membrane 1 Modal Analysis x y (1; 0) T (1; 0) T (0; 1) T (0;1) T a b Figure 1: A rectangular membrane in cartesian coordinates Consider a rectangular membrane with all edges xed as shown in Fig. 1, and governed by w ;tt T(w ;xx + w ;yy ) = 0; (1) w(0; y; t) = 0; w(a; y; t) = 0; w(x; 0; t) = 0; w(x; b; t) = 0: (2) Assume a solution of the form w(x; y; t) = W(x; y)e i!t ; (3) where W(x; y) is the eigenfunction, and ! is the circular eigenfrequency. Substituting this solution into the equation of motion leads to the eigenvalue 2 problem of the Helmholz equation r 2 W + ! 2 c 2 W = 0 (4) W(0; y) = 0; W(a; y) = 0; W(x; 0) = 0; and W(x; b) = 0 (5) whererW = W ;xx + W ;yy . Assume a solution of (4) of the form W(x; y) = Be i(k x x+k y y) ; (6) where B is a complex constant. Substituting (6) in (4) yields the dispersion relation of the membrane as k 2 x k 2 y + ! 2 c 2 = 0: (7) The solutions of (7) are of the form k x = and k y = such that 2 + 2 = ! 2 =c 2 . The general solution of (4) can then be written as W(x; y) = (B 1 e ix + B 2 e ix )(B 3 e iy + B 4 e iy ); (8) or in the form W(x; y) = A 1 cos x cosy + A 2 cos x siny + A 3 sin x cosy + A 4 sin x siny: (9) Using (9) in the rst and third boundary conditions in (5), one obtains A 1 cos y + A 2 sin y = 0 (10) and A 1 cos x + A 3 sin x = 0: (11) 3 This implies A 1 = 0, A 2 = 0 and A 3 = 0, and hence W(x; y) = A 4 sinx sin y: (12) The second and fourth boundary conditions in (5) now lead to sin a sin y = 0 and sinx sin b = 0 (13) which requires = m a ; and = n b ; m; n = 1; 2; : : :;1: (14) The eigenfunctions are then obtained from (12) as W (m;n) = sin mx a sin nx b ; m; n = 1; 2; : : :;1; (15) where W (m;n) represents the eigenfunction of the (m; n) mode. These eigen- functions satisfy the orthogonality condition hW (m;n) (x; y); W (r;s) (x; y)i = Z a 0 Z b 0 W (m;n) (x; y)W (r;s) (x; y) dx dy = ab 4 mr ns : (16) The rst few mode-shapes of the membrane are shown in Fig. 2. Using (14) in the condition 2 + 2 = ! 2 =c 2 yields the frequency equation ! (m;n) = c r m 2 a 2 + n 2 b 2 ; (17) where ! (m;n) represents the circular eigenfrequency of the (m; n) mode. Note that the eigenfrequencies of the membrane are not, in general, integral mul- tiples of the fundamental frequency (as in the case of a string). 4 m = 1; n = 1 m = 1; n = 2 m = 2; n = 1 m = 2; n = 2 m = 3; n = 1 m = 3; n = 2 Figure 2: First few mode shapes of a rectangular membrane with xed boundaries 5Read More

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