# Free Vibration Analysis of Rectangular Orthotropic Membranes in Large Deflection

Authors: Zheng Zhou-Lian; Liu Chang-Jiang; He Xiao-Ting; Chen Shan-Lin
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

This paper reviewed the research on the vibration of orthotropic membrane, which commonly applied in the membrane structural engineering. We applied the large deflection theory of membrane to derive the governing vibration equations of orthotropic membrane, solved it, and obtained the power series formula of nonlinear vibration frequency of rectangular membrane with four edges fixed. The paper gave the computational example and compared the two results from the large deflection theory and the small one, respectively. Results obtained from this paper provide some theoretical foundation for the measurement of pretension by frequency method; meanwhile, the results provide some theoretical foundation for the research of nonlinear vibration of membrane structures and the response solving of membrane structures under dynamic loads.


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## Body

## 1.  Introduction

The membrane structure is a new structure system, which has been rising  during  the recent
several dozens  of  years. Because of its economy, beauty, and less dead weight,  it is widely applied
to large span structures,  such as large-scale stadium,  exhibition center, works of decoration,
and so on. The application of tensile membrane  structure was the most extensive in each kind of
membrane structure  systems.  The tensile membrane structure's stiffness is formed by the zonal
and meridional  pretensions.  The tallying degree of the actual pretension value and the design
pretension value directly  influences  the normal utilization and safety after the construction
finished [1, 2]. Therefore, the
nondestructive monitoring of the membrane structure's pretension is very important and necessary
after the construction finished.At present, most researches focused on design and construction of membrane structures; however,
there is little research on the pretension measurement of membrane structures
[1, 2]. The main methods of the
pretension measurement include strain method, frequency method, deflection method, and “cable
analogy” method [3]. If we study the application of frequency
method, the vibration theory of membrane must be  involved.  Many scholars studied about the
vibration theory of membrane. Their researches involve the problem of  free vibration of a confocal
composite elliptical membrane [4], the problem of fundamental
frequency of rectangular membranes with an internal oblique support
[5], the problem of free vibration of composite rectangular
membranes with an oblique and a bent interface [6, 7], the problem of free in-plane vibration of an axially moving
membrane [8]. However, these researches did not  aim  at this kind
of orthotropic membrane that  was  used in construction field. Moreover, these researches have not
obtained the power series formula of nonlinear vibration frequency of the orthotropic membrane.In this paper, we studied the vibration of orthotropic membranes according to the large
deflection  theory of membrane   [9, 10] and obtained the power series formula of nonlinear vibration
frequency of rectangular membranes with four edges fixed. The paper gave the computational example
and compared the two results from the large deflection theory and the small one, respectively.


## 2. Governing Equations and Boundary Conditions

The studied rectangular membrane is orthotropic. Its two orthogonal directions are the two
principal fiber directions, and the material characteristics of the two principal fiber directions
are different. Assume that the studied rectangular membrane is fixed on four edges. The two
principal fiber directions arex and y, respectively. a and b denote the length of x and y direction, respectively; N0x and N0y denote initial tension in x and y, respectively, as shown in Figure 1.Figure 1
Rectangular  membrane with four edges fixed.According to the large deflection theory and D'Alembert's principle of membranes [8–10], the vibration partial differential equation and consistency equation of orthotropic membrane are (2.1)ρ∂2w∂t2−(Nx+N0x)∂2w∂x2−(Ny+N0y)∂2w∂y2=01E1h∂2Nx∂y2−μ2E2h∂2Ny∂y2−μ1E1h∂2Nx∂x2+1E2h∂2Ny∂x2−1Gh∂2Nxy∂x∂y=(∂2w∂x∂y)2−∂2w∂x2∂2w∂y2, where ρ denotes aerial density of membrane; Nx and Ny denote tension in x and y, respectively; N0x and N0y denote initial tension in x and y, respectively; Nxy denotes shear force; w denotes deflection: w(x,y,t);  h denotes membrane's thickness; E1 and E2 denote Young's modulus in x and y, respectively; G denotes shearing modulus; μ1 and μ2 denote Poisson's ratio in x and y, respectively.While the membrane is in vibration, the effect of shearing stress is so small that we may takeNxy=0. Introducing the stress function and letting Nx=h(∂2φ/∂y2), Ny=h(∂2φ/∂x2), N0x=h⋅σ0x, and N0y=h⋅σ0y, (2.1) can be simplified as follows: (2.2)ρh∂2w∂t2−(σ0x+∂2φ∂y2)∂2w∂x2−(σ0y+∂2φ∂x2)∂2w∂y2=0,(2.3)1E1∂4φ∂y4+1E2∂2φ∂x4=(∂2w∂x∂y)2−∂2w∂x2∂2w∂y2, where φ denotes stress function: φ(x,y,t); σ0x and σ0y denote initial tensile stress in x and y, respectively.The corresponding boundary conditions can be expressed as follows:(2.4)w(0,y,t)=0,w(a,y,t)=0,w(x,0,t)=0,w(x,b,t)=0.


## 3. Solution of Free Vibration Frequency

Functions that satisfy the boundary conditions (2.4) are taken as follows: (3.1)w(x,y,t)=W(x,y)T(t),φ(x,y,t)=ϕ(x,y)T2(t), where W(x,y) is the given mode shape function, and ϕ(x,y) and T(t) are the unknown functions.Assume that the mode shape function is as follows:(3.2)W(x,y)=sinmπxasinnπyb, where m and n are integers and denote the sine half-wave number in x and y, respectively. Equation (3.2) satisfies the boundary conditions automatically.Substituting (3.1) into (2.3) yields (3.3)1E1∂4ϕ∂y4+1E2∂2ϕ∂x4=(∂2W∂x∂y)2−∂2W∂x2∂2W∂y2.Substituting (3.2) into (3.3) yields (3.4)1E1∂4ϕ∂y4+1E2∂2ϕ∂x4=m2n2π42a2b2(cos2mπxa+cos2nπyb).Assume that the solution of (3.4) is (3.5)ϕ(x,y)=α⋅cos2mπxa+β⋅cos2nπyb.Substituting (3.5) into (3.4) yields (3.6)α=E2n2a232m2b2,β=E1m2b232n2a2.Substituting (3.1) into (2.2), according to the Galerkin method [8, 10], yields (3.7)∬S(ρh∂2w∂t2−∂2φ∂y2∂2w∂x2−∂2φ∂x2∂2w∂y2)W(x,y)ds=∬S[ρhW∂2T(t)∂t2−(σ0x⋅∂2W∂x2+σ0y⋅∂2W∂y2)T(t)(∂2ϕ∂y2∂2W∂x2+∂2ϕ∂x2∂2W∂y2)T3(t)]W(x,y)ds=0.Obviously, (3.7) is a nonlinear differential equation with respect to T(t), and it can be expressed as follows: (3.8)A⋅d2T(t)dt2−B⋅T(t)−C⋅T3(t)=0, where
(3.9)A=∬SρhW2ds=∬Sρhsin2mπxasin2nπybds=ρab4h,B=∬S(σ0x⋅∂2W∂x2+σ0y⋅∂2W∂y2)Wds=−π2ab4(m2a2σ0x+n2b2σ0y),C=∬S(∂2ϕ∂y2∂2W∂x2+∂2ϕ∂x2∂2W∂y2)Wds=−abπ464(E1m4a4+E2n4b4).Substituting the values ofA, B, and C into (3.8) and dividing by ρab/4h on two ends yields (3.10)d2T(t)dt2+hπ2ρ(m2a2σ0x+n2b2σ0y)T(t)+hπ416ρ(E1m4a4+E2n4b4)T3(t)=0.LettingM=(hπ2/ρ)((m2/a2)σ0x+(n2/b2)σ0y), N=(π4h/16ρ)((E1m4/a4)+(E2n4/b4)) yields (3.11)d2T(t)dt2+M⋅T(t)+N⋅T3(t)=0.Integrating (3.11) yields (3.12)(dT(t)dt)2+M⋅T2(t)+N2⋅T4(t)=H.In (3.12), the value of H is determined by the initial conditions. Assume that the initial displacement is T|t=0=T0. T0 is the amplitude of the membrane, so the initial velocity is (3.13)dT(t)dt|t=0=0.SubstitutingT|t=0=T0 and (3.13) into (3.12) yields H=MT02+(N/2)T04. Then substituting H=MT02+(N/2)T04 into (3.12) yields (3.14)dT(t)dt=T02MT02+N2⋅(1−T2T02)⋅(1+NT022M+NT02⋅T2T02), letting λ=T02M/T02+N/2, k2=NT02/(2M+NT02).Integrating (3.11) by the separate variable method, we can obtain the period of the vibration of the membrane: (3.15)Z=4λ∫0T0[(1−T2T02)(1+k2⋅T2T02)]−1/2dT.LettingT/T0=sinθ, (3.15) can be transformed into (3.16)Z=4T0λ∫0π/2(1+k2⋅sin2θ)−1/2dθ(0≤k⋅sinθ≤1),  where  (1+k2⋅sin2θ)−1/2 may be spread as a power series with respect to k⋅sinθ: (3.17)(1+k2⋅sin2θ)−1/2=1−12k2sin2θ+1⋅32⋅4k4sin4θ+⋯(−1)n(2p−1)!!(2p)!!(ksinθ)2p.Substituting (3.17) into (3.16) then  solving  (3.16) through integrating item by item [10, 11] yields (3.18)Z=4T0λ∫0π/2[1−12k2sin2θ+1⋅32⋅4k4sin4θ+⋯(−1)n(2p−1)!!(2p)!!(ksinθ)2p]dθ=2πT0λ[1−(12)2k2+(1⋅32⋅4)2k2+⋯+(−1)p((2p−1)!!(2p)!!)2k2p]=2πT0λ∑n=0∞(−1)p((2p−1)!!(2p)!!)2k2p, where p=0,1,2,3,….Therefore, the vibration frequency of the membrane is(3.19)ω=2πZ=M+(N/2)T02∑n=0∞(−1)p((2p−1)!!/(2p)!!)2(NT02/(2M+NT02))p, where M=(hπ2/ρ)((m2/a2)σ0x+(n2/b2)σ0y), N=(π4h/16ρ)(E1m4/a4+E2n4/b4), and T0 is the amplitude of the membrane.In (3.19), letting T0→0, (3.19) can be transformed into the formula obtained according to the small deflection: (3.20)ω=πhρ⋅σ0x⋅m2a2+σ0y⋅n2b2.


## 4. Computational Examples and Discussion

Take the membrane material commonly applied in project as an example. Young's  moduli  inx and y are E1=1.4×106KN/m2 and E2=0.9×106KN/m2, respectively; the aerial density of membranes is ρ=1.7kg/m2;  the membrane's thickness is h=1.0mm.  Calculate  the vibration frequency of the membrane according to (3.19).We can draw the conclusion from the result of Table1. If we exchange the two lengths of the two orthogonal directions and consider the orthotropic characteristic of the membrane, the result is dissimilar with the one before the exchange; however, if we only exchange the two lengths of the two orthogonal directions and do not consider the orthotropic characteristic, the result  is  similar with the one before the exchange. Therefore, we need to consider the orthotropic characteristic of membranes in practical engineering.Table 1
T0=0.1m, σ0x=σ0y=5.0×103KN/m2, and m=n=1.
Ma=1,a=1,a=2,a=1,a=3,a=1,a=4,a=1,a=5,b=1b=2b=1b=3b=1b=4b=1b=5b=1Rad/s344.231273.032249.679245.268237.578260.131233.909258.752232.316Hz54.78643.45539.73839.03637.81241.40137.22741.18236.974We can draw the conclusion from the result of Table2. The initial displacement (the amplitude) has influenced the vibration frequency of the rectangle membrane when  calculated  according to the large deflection theory. The frequency enlarged with the increase of the initial displacement, and the larger the initial displacement is, the larger the effect on the frequency is and vice versa. When the initial displacement  approaches  zero, the result is consistent with that obtained according to the small deflection theory.Table 2
a=1m, b=1m, σ0x=σ0y=5.0×103KN/m2, and m=n=1.
MT0= 0.2T0= 0.15T0= 0.10T0= 0.05T0= 0.01T0= 0.005T0= 0.001T0= 0.0001T0→0Rad/s544.651439.21344.231270.86242.227286.325240.962240.949240.948Hz86.68469.90254.78643.10938.55245.57038.35038.34838.348According to (3.20) (the frequency formula of the small deflection theory),  by  calculating the vibration frequency of the membrane, the results are listed in Table 6.The results from Table3 to Table 6 are plotted as shown in Figure 2.Table 3
a=1m, b=1m, σ0x=σ0y=5.0×103KN/m2, and T0=0.10m.
Orderm=1, n=1m=1, n=2m=2, n=1m=2, n=2m=1, n=3m=3, n=1m=3, n=3Rad/s344.231745.731865.6411089.301486.921799.042308.23Hz54.786118.687137.771173.368236.650286.326367.367Table 4
a=1m, b=1m, σ0x=σ0y=5.0×103KN/m2, and T0=0.05m.
Orderm=1, n=1m=1, n=2m=2, n=1m=2, n=2m=1, n=3m=3, n=1m=3, n=3Rad/s270.86499.358546.065688.462880.8461016.961317.63Hz43.10979.47586.909109.572140.191161.854209.707Table 5
a=1m, b=1m, σ0x=σ0y=5.0×103KN/m2, and T0=0.01m.
Orderm=1, n=1m=1, n=2m=2, n=1m=2, n=2m=1, n=3m=3, n=1m=3, n=3Rad/s242.227386.497389.087492.030556.943566.445756.548Hz38.55261.51361.92578.30988.64090.153120.408Table 6
a=1m, b=1m, and σ0x=σ0y=5.0×103KN/m2.
Orderm=1, n=1m=1, n=2m=2, n=1m=2, n=2m=1, n=3m=3, n=1m=3, n=3Rad/s240.949380.974380.974481.898538.779538.779722.847Hz38.34860.63460.63476.69785.74985.749115.045Figure 2
Comparative analysis of the results based on the large deflection theory and the small one.We can draw conclusions from the analysis of Figure2. While order keeps unchanged, all frequency results based on the large deflection theory are larger than the corresponding ones based on the small deflection theory. The larger the initial displacement is, the larger the frequency is and vice versa. Specially, the smaller the initial displacement is, the closer the two results based on large and small deflection theories are.The above conclusions are analyzed as follows.(i)
We considered the rigidity change caused by geometrical large deflection of membrane when calculating the frequency according to the large deflection theory. When the lateral displacement of membrane increased, the inner force increased, and the lateral rigidity also increased; then the vibration was quickened. Therefore, the membrane's vibration frequency will enlarge with the increase of initial displacement.(ii)
When initial displacement of membrane is very little, the change of lateral rigidity is also very little in the process of vibration, so it is negligible. In this case, the computational result based on the large deflection theory is very close to that based on the small deflection theory, which considers no changes of the lateral rigidity in the process of vibration.


## 5. Conclusions

(i)
We applied the large deflection theory of membranes and  D'Alembert's  principle derived the governing vibration equations of rectangular membrane with four edges fixed, solved it, and obtained the power series formula of nonlinear vibration frequency of rectangular membranes with four edges fixed.(ii)
The frequency formula obtained in this paper is dependent on the initial conditions,  as  we considered the change of the lateral rigidity in the process of  vibration. Therefore, the formula has reflected the geometric nonlinear characteristic of membrane structure's vibration. This is more tally with the actual situation and more reasonable than the result that is calculated according to the small deflection theory.(iii)
The nonlinear governing equation and the power series formula obtained in this paper provide some theoretical foundation for the measurement of pretension by frequency method; meanwhile, the results provide some theoretical foundation for the research of nonlinear vibration of membrane structures and the response solving of membrane structures under dynamic loads.



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*Source: 634362-2009-04-06.xml*