Free Vibration Analysis of Rectangular Orthotropic Membranes in Large Deflection

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.


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.

Mathematical Problems in Engineering
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.

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 are x and y, respectively.a and b denote the length of x and y direction, respectively; N 0x and N 0y denote initial tension in x and y, respectively, as shown in Figure 1.
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 where ρ denotes aerial density of membrane; N x and N y denote tension in x and y, respectively; N 0x and N 0y denote initial tension in x and y, respectively; N xy denotes shear force; w denotes deflection: w x, y, t ; h denotes membrane's thickness; E 1 and E 2 denote Young's modulus in x and y, respectively; G denotes shearing modulus; μ 1 and μ 2 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 take N xy 0. Introducing the stress function and letting N 0x h•σ 0x , and N 0y h•σ 0y , 2.1 can be simplified as follows: where ϕ denotes stress function: ϕ x, y, t ; σ 0x and σ 0y denote initial tensile stress in x and y, respectively.
The corresponding boundary conditions can be expressed as follows: w 0, y, t 0, w a, y, t 0, w x, 0, t 0, w x, b, t 0. 2.4

Solution of Free Vibration Frequency
Functions that satisfy the boundary conditions 2.4 are taken as follows: 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: 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.
Mathematical Problems in Engineering Substituting 3.1 into 2.3 yields Assume that the solution of 3.4 is Substituting 3.1 into 2.2 , according to the Galerkin method 8, 10 , yields Obviously, 3.7 is a nonlinear differential equation with respect to T t , and it can be expressed as follows: where

3.9
Substituting the values of A, B, and C into 3.8 and dividing by ρab/4h on two ends yields Integrating 3.11 yields

3.12
In 3.12 , the value of H is determined by the initial conditions.Assume that the initial displacement is T | t 0 T 0 .T 0 is the amplitude of the membrane, so the initial velocity is dT t dt t 0 0.

3.13
Substituting T | t 0 T 0 and 3.13 into 3.12 yields H MT 2 0 N/2 T 4 0 .Then substituting H MT 2 0 N/2 T 4 0 into 3.12 yields 11 by the separate variable method, we can obtain the period of the vibration of the membrane: dT.

3.15
Letting T/T 0 sin θ, 3.15 can be transformed into where 1 k 2 •sin 2 θ −1/2 may be spread as a power series with respect to k• sin θ: 3.17 Substituting 3.17 into 3.16 then solving 3.16 through integrating item by item 10, 11 yields
In 3.19 , letting T 0 → 0, 3.19 can be transformed into the formula obtained according to the small deflection: 3.20

Computational Examples and Discussion
Take the membrane material commonly applied in project as an example.Young's moduli in x and y are E 1 1.4 × 10 6 KN/m 2 and E 2 0.9 × 10 6 KN/m 2 , respectively; the aerial density of membranes is ρ 1.7 kg/m 2 ; the membrane's thickness is h 1.0 mm.Calculate the vibration frequency of the membrane according to 3.19 .We can draw the conclusion from the result of Table 1.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.
We can draw the conclusion from the result of Table 2.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.
The results from Table 3 to Table 6 are plotted as shown in Figure 2.
We can draw conclusions from the analysis of Figure 2.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.

Mathematical Problems in Engineering
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.

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.
According 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.

Figure 1 :
Figure 1: Rectangular membrane with four edges fixed.

Figure 2 :
Figure 2: Comparative analysis of the results based on the large deflection theory and the small one.