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
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Rectangular Membrane - Vibration of Structures - Lecture Notes, Study notes of Structural Design and Architecture
Some basic concept Vibration of Structures are Harmonic Waves, Influence of Axial Force, Initial Value Problem, Mathematical Modeling, Modal Analysis, Motion of Material Points, Orthogonality Relations, Projection Methods.Main points of this lecture are: Rectangular Membrane, Modal Analysis, Modal Degeneracy, Membrane Vibrations, Degeneracy, Helmholz Equation, Mode-Shapes of Membrane, Frequency Equation, Free Vibration Problem, Arbitrary Linear Combination
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2012/2013
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Vibrations of Structures
Module IV: Vibrations of Membranes
Lesson 32: The Rectangular Membrane
Contents:
- Modal Analysis
- 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 fixed as shown in Fig. 1, and governed by
μw,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)eiωt, (3)
where W (x, y) is the eigenfunction, and ω is the circular eigenfrequency. Substituting this solution into the equation of motion leads to the eigenvalue 2
This implies A 1 = 0, A 2 = 0 and A 3 = 0, and hence
W (x, y) = A 4 sin αx sin βy. (12)
The second and fourth boundary conditions in (5) now lead to
sin αa sin βy = 0 and sin αx sin βb = 0 (13)
which requires
α = mπ a , and β = nπ b , m, n = 1, 2 ,... , ∞. (14)
The eigenfunctions are then obtained from (12) as
W (m,n)^ = sin mπx a sin nπx b , m, n = 1, 2 ,... , ∞, (15)
where W (m,n)^ represents the eigenfunction of the (m, n) mode. These eigen- functions satisfy the orthogonality condition
〈W (m,n)(x, y), W (r,s)(x, y)〉 =
∫ (^) a 0
∫ (^) b 0 W^
(m,n)(x, y)W (r,s)(x, y) dx dy = ab 4 δmrδns. (16)
The first few mode-shapes of the membrane are shown in Fig. 2. Using (14) in the condition α^2 + β^2 = ω^2 /c^2 yields the frequency equation
ω(m,n) = πc
√m 2 a^2 +^
n^2 b^2 ,^ (17) where ω(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 fixed boundaries
5
δ = 0 δ = π/ 25
δ = π/ 10 δ = π/ 4
Figure 3: Nodal curves of linear combinations of the degenerate modes (3,1) and (1,3) of asquare membrane
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