Free Vibration Analysis of Rectangular Orthotropic Membranes in Large Deflection

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.

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 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.

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 𝜌𝜕²𝑤𝜕𝑡²−(𝑁_𝑥+𝑁_0𝑥)𝜕²𝑤𝜕𝑥²−(𝑁_𝑦+𝑁_0𝑦)𝜕²𝑤𝜕𝑦²1=0𝐸₁ℎ𝜕²𝑁_𝑥𝜕𝑦²−𝜇₂𝐸₂ℎ𝜕²𝑁_𝑦𝜕𝑦²−𝜇₁𝐸₁ℎ𝜕²𝑁_𝑥𝜕𝑥²+1𝐸₂ℎ𝜕²𝑁_𝑦𝜕𝑥²−1𝜕𝐺ℎ²𝑁_𝑥𝑦𝜕𝜕𝑥𝜕𝑦=(²𝑤)𝜕𝑥𝜕𝑦²−𝜕²𝑤𝜕𝑥²𝜕²𝑤𝜕𝑦², (2.1) 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; 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 take 𝑁_𝑥𝑦=0. Introducing the stress function and letting 𝑁_𝑥=ℎ(𝜕²𝜑/𝜕𝑦²), 𝑁_𝑦=ℎ(𝜕²𝜑/𝜕𝑥²), 𝑁_0𝑥=ℎ⋅𝜎_0𝑥, and 𝑁_0𝑦=ℎ⋅𝜎_0𝑦, (2.1) can be simplified as follows: 𝜌ℎ𝜕²𝑤𝜕𝑡²−(𝜎_0𝑥+𝜕²𝜑𝜕𝑦²)𝜕²𝑤𝜕𝑥²−(𝜎_0𝑦+𝜕²𝜑𝜕𝑥²)𝜕²𝑤𝜕𝑦²1=0,𝐸₁𝜕⁴𝜑𝜕𝑦⁴+1𝐸₂𝜕²𝜑𝜕𝑥⁴𝜕=(²𝑤)𝜕𝑥𝜕𝑦²−𝜕²𝑤𝜕𝑥²𝜕²𝑤𝜕𝑦², (2.2)(2.3) 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)

3. Solution of Free Vibration Frequency

Functions that satisfy the boundary conditions (2.4) are taken as follows: 𝑤(𝑥,𝑦,𝑡)=𝑊(𝑥,𝑦)𝑇(𝑡),𝜑(𝑥,𝑦,𝑡)=𝜙(𝑥,𝑦)𝑇²(𝑡), (3.1) 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: 𝑊(𝑥,𝑦)=sin𝑚𝜋𝑥𝑎sin𝑛𝜋𝑦𝑏, (3.2) 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 1𝐸₁𝜕⁴𝜙𝜕𝑦⁴+1𝐸₂𝜕²𝜙𝜕𝑥⁴𝜕=(²𝑊)𝜕𝑥𝜕𝑦²−𝜕²𝑊𝜕𝑥²𝜕²𝑊𝜕𝑦². (3.3)

Substituting (3.2) into (3.3) yields 1𝐸₁𝜕⁴𝜙𝜕𝑦⁴+1𝐸₂𝜕²𝜙𝜕𝑥⁴=𝑚²𝑛²𝜋⁴2𝑎²𝑏²(cos2𝑚𝜋𝑥𝑎+cos2𝑛𝜋𝑦𝑏). (3.4)

Assume that the solution of (3.4) is 𝜙(𝑥,𝑦)=𝛼⋅cos2𝑚𝜋𝑥𝑎+𝛽⋅cos2𝑛𝜋𝑦𝑏. (3.5)

Substituting (3.5) into (3.4) yields 𝐸𝛼=²𝑛²𝑎²32𝑚²𝑏²𝐸,𝛽=¹𝑚²𝑏²32𝑛²𝑎². (3.6)

Substituting (3.1) into (2.2), according to the Galerkin method [8, 10], yields _𝑆(𝜌ℎ𝜕²𝑤𝜕𝑡²−𝜕²𝜑𝜕𝑦²𝜕²𝑤𝜕𝑥²−𝜕²𝜑𝜕𝑥²𝜕²𝑤𝜕𝑦²=)𝑊(𝑥,𝑦)𝑑𝑠_𝑆[𝜌ℎ𝑊𝜕²𝑇(𝑡)𝜕𝑡²−(𝜎_0𝑥⋅𝜕²𝑊𝜕𝑥²+𝜎_0𝑦⋅𝜕²𝑊𝜕𝑦²𝜕)𝑇(𝑡)(²𝜙𝜕𝑦²𝜕²𝑊𝜕𝑥²+𝜕²𝜙𝜕𝑥²𝜕²𝑊𝜕𝑦²)𝑇³(𝑡)]𝑊(𝑥,𝑦)𝑑𝑠=0. (3.7)

Obviously, (3.7) is a nonlinear differential equation with respect to 𝑇(𝑡), and it can be expressed as follows: 𝑑𝐴⋅²𝑇(𝑡)𝑑𝑡²−𝐵⋅𝑇(𝑡)−𝐶⋅𝑇³(𝑡)=0, (3.8) where 𝐴=_𝑆𝜌ℎ𝑊²𝑑𝑠=_𝑆𝜌ℎsin²𝑚𝜋𝑥𝑎sin²𝑛𝜋𝑦𝑏𝑑𝑠=𝜌𝑎𝑏,4ℎ𝐵=_𝑆(𝜎_0𝑥⋅𝜕²𝑊𝜕𝑥²+𝜎_0𝑦⋅𝜕²𝑊𝜕𝑦²𝜋)𝑊𝑑𝑠=−²𝑎𝑏4(𝑚²𝑎²𝜎_0𝑥+𝑛²𝑏²𝜎_0𝑦),𝐶=_𝑆(𝜕²𝜙𝜕𝑦²𝜕²𝑊𝜕𝑥²+𝜕²𝜙𝜕𝑥²𝜕²𝑊𝜕𝑦²)𝑊𝑑𝑠=−𝑎𝑏𝜋⁴(𝐸64¹𝑚⁴𝑎⁴+𝐸₂𝑛⁴𝑏⁴). (3.9)

Substituting the values of 𝐴, 𝐵, and 𝐶 into (3.8) and dividing by 𝜌𝑎𝑏/4ℎ on two ends yields 𝑑²𝑇(𝑡)𝑑𝑡²+ℎ𝜋²𝜌(𝑚²𝑎²𝜎_0𝑥+𝑛²𝑏²𝜎_0𝑦)𝑇(𝑡)+ℎ𝜋⁴(𝐸16𝜌¹𝑚⁴𝑎⁴+𝐸₂𝑛⁴𝑏⁴)𝑇³(𝑡)=0. (3.10)

Letting 𝑀=(ℎ𝜋²/𝜌)((𝑚²/𝑎²)𝜎_0𝑥+(𝑛²/𝑏²)𝜎_0𝑦), 𝑁=(𝜋⁴ℎ/16𝜌)((𝐸₁𝑚⁴/𝑎⁴)+(𝐸₂𝑛⁴/𝑏⁴)) yields 𝑑²𝑇(𝑡)𝑑𝑡²+𝑀⋅𝑇(𝑡)+𝑁⋅𝑇³(𝑡)=0. (3.11)

Integrating (3.11) yields (𝑑𝑇(𝑡))𝑑𝑡²+𝑀⋅𝑇²𝑁(𝑡)+2⋅𝑇⁴(𝑡)=𝐻. (3.12)

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 𝑑𝑇(𝑡)|𝑑𝑡_𝑡=0=0. (3.13)

Substituting 𝑇|_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 𝑑𝑇(𝑡)𝑑𝑡=𝑇₀²𝑀𝑇₀²+𝑁2⋅𝑇(1−²𝑇₀²)⋅(1+𝑁𝑇₀²2𝑀+𝑁𝑇₀²⋅𝑇²𝑇₀²), (3.14) 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: 4𝑍=𝜆^𝑇0₀𝑇[(1−²𝑇₀²)(1+𝑘²⋅𝑇²𝑇₀²)]^−1/2𝑑𝑇. (3.15)

Letting 𝑇/𝑇₀=sin𝜃, (3.15) can be transformed into 𝑍=4𝑇₀𝜆₀^𝜋/2(1+𝑘²⋅sin²𝜃)^−1/2𝑑𝜃(0≤𝑘⋅sin𝜃≤1), (3.16) where (1+𝑘²⋅sin²𝜃)^−1/2 may be spread as a power series with respect to 𝑘⋅sin𝜃: (1+𝑘²⋅sin²𝜃)^−1/21=1−2𝑘²sin²𝜃+1⋅3𝑘2⋅4⁴sin⁴𝜃+⋯(−1)^𝑛(2𝑝−1)!!(2𝑝)!!(𝑘sin𝜃)^2𝑝. (3.17)

Substituting (3.17) into (3.16) then solving (3.16) through integrating item by item [10, 11] yields 𝑍=4𝑇₀𝜆₀^𝜋/21[1−2𝑘²sin²𝜃+1⋅3𝑘2⋅4⁴sin⁴𝜃+⋯(−1)^𝑛(2𝑝−1)!!(2𝑝)!!(𝑘sin𝜃)^2𝑝=]𝑑𝜃2𝜋𝑇₀𝜆1[1−(2)²𝑘²+(1⋅3)2⋅4²𝑘²+⋯+(−1)^𝑝((2𝑝−1)!!)(2𝑝)!!²𝑘^2𝑝]=2𝜋𝑇₀𝜆^∞_𝑛=0(−1)^𝑝((2𝑝−1)!!)(2𝑝)!!²𝑘^2𝑝, (3.18) where 𝑝=0,1,2,3,….

Therefore, the vibration frequency of the membrane is 𝜔=2𝜋𝑍=𝑀+(𝑁/2)𝑇₀²∑^∞_𝑛=0(−1)^𝑝((2𝑝−1)!!/(2𝑝)!!)²(𝑁𝑇₀²/(2𝑀+𝑁𝑇₀²))^𝑝, (3.19) where 𝑀=(ℎ𝜋²/𝜌)((𝑚²/𝑎²)𝜎_0𝑥+(𝑛²/𝑏²)𝜎_0𝑦), 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: 𝜔=𝜋ℎ𝜌⋅𝜎_0𝑥⋅𝑚²𝑎²+𝜎_0𝑦⋅𝑛²𝑏². (3.20)

4. Computational Examples and Discussion

Take the membrane material commonly applied in project as an example. Young's moduli in 𝑥 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 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.

Table 1 

𝑇₀=0.1m, 𝜎_0𝑥=𝜎_0𝑦=5.0×10³KN/m², and 𝑚=𝑛=1.

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.

Table 2 

𝑎=1m, 𝑏=1m, 𝜎_0𝑥=𝜎_0𝑦=5.0×10³KN/m², and 𝑚=𝑛=1.

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.

The results from Table 3 to Table 6 are plotted as shown in Figure 2.

Table 3 

𝑎=1m, 𝑏=1m, 𝜎_0𝑥=𝜎_0𝑦=5.0×10³KN/m², and 𝑇₀=0.10m.

Table 4 

𝑎=1m, 𝑏=1m, 𝜎_0𝑥=𝜎_0𝑦=5.0×10³KN/m², and 𝑇₀=0.05m.

Table 5 

𝑎=1m, 𝑏=1m, 𝜎_0𝑥=𝜎_0𝑦=5.0×10³KN/m², and 𝑇₀=0.01m.

Table 6 

𝑎=1m, 𝑏=1m, and 𝜎_0𝑥=𝜎_0𝑦=5.0×10³KN/m².

Figure 2 

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

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. (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.

Acknowledgment

This work is supported by Chongqing Municipal Construction Committee, Construction and Scientific 2008, Project no. 73.