Stability of membrane in low subsonic flow

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Abstract

Stability of an isolated membrane lying in a uniform two-dimensional low subsonic flow is studied theoretically and experimentally. The problem is formulated in a form of a boundary integral equation and differential equations. The boundary integral equation is solved by the boundary element method and the finite difference method is used to solve the differential equations. An effect of a membrane wake is used in the analysis. The theoretical critical divergence velocity is compared with the experimental value.

Introduction

Stability of membranes in a flow has a practical meaning in the case of membrane roofs made of technical fabric used in civil engineering. At significant values of an air flow velocity divergence and flutter type of stability loss may occur. These problems were theoretically and experimentally analyzed by many authors (see, e.g., [1], [3], [4], [5]).

In problems of structures in a fluid flow non-conservative loading takes place. Several papers were devoted to such an analysis. The influence of damping is especially important in the behavior of such systems (see [7], [8], [9]).

Experiments in a wind tunnel carried out by the author on membranes made of latex rubber showed that after the divergent loss of stability large deflections occur in membranes and subsequent phenomena are non-linear. It was observed, that the divergent mode was not symmetric with respect to the centre line (x=l/2). These experiments were a motivation to create a theoretical model, which would allow to obtain the mode similar to the one observed in the experiment.

In the theoretical model a membrane supported at two edges lying in a two-dimensional subsonic flow is analyzed. The membrane wake formed behind the trailing edge was taken into account. The problem was described by one boundary integral equation and three differential equations. The boundary element method and the finite difference method were used in the solution.

Section snippets

Experimental investigations

Experimental testing of a membrane with length l=535mm and width b=75mm was carried out. The membrane was made of two 0.8 mm thick layers of latex rubber. It was simply supported on two edges. The initial tension was T=288N/m. Fig. 1 presents the sketch of the model. The wind tunnel, where the membrane stability tests were carried out is a closed circuit tunnel with a 920×640mm closed test section. The membrane behavior was monitored by a video camera.

It was observed that a limited increase of

Theoretical analysis

The one-dimensional equation of motion of a membrane shown in Fig. 4 subjected to a fluid flow on both sides is given by-T2wx2+μ2wt2+Δp=0,where T is the tension in the membrane per unit length, μ—the membrane mass per unit area, ww(x,t)—its lateral deflection, ΔpΔp(x,t)—the load per unit area on the membrane equal to the difference between the perturbation pressures on the upper and lower surfaces of the membrane caused by its deflection (Δp=p1-p2,|p1|=|p2|). The leading and trailing

Numerical solution of the problem

The problem was described by one boundary integral equation (5), three differential equations (1), (2) and (6) as well as by the wake condition (7). Upon separation of the space and time variables and expressing the solution with respect to time in the exponential form (e.g., Δp(x,t)=Δp˜(x)eλt, λ is the eigenvalue parameter, λ=iω, i=-1, ω is the complex eigenfrequency (ω=ωR+iωI)), Eqs. (1), (2), (5) and (6) yield-T2w˜(x)x2+λμw˜(x)+Δp˜(x)=0,Δp˜(x)=-2ρλΦ˜(x)+UΦ˜(x)x,v˜z(x)=-1πSΦ˜(x)2z2(ln(

Numerical results

Computer programs were written based on the formulation presented in Sections 3 and 4. The dimensionless parameters used in the analysis are U2¯=U2ρl/T, ω2¯=ω2l2μ/T and μ¯=μ̲/(ρl), where U¯ is the dimensionless flow velocity, ω¯ is the dimensionless frequency and μ¯ is the dimensionless mass of the membrane.

The divergence velocity of the membrane was calculated using Eq. (24). Fifty collocation points on the membrane and on its wake were used. The calculations were performed with and without

Conclusions

Results of theoretical and experimental analysis of stability of a membrane in a two-dimensional subsonic flow are presented in this paper. The influence of membrane wake on its stability was investigated. The problem related to membranes has not been analyzed yet. The boundary element method and the finite difference method were used in the numerical solution of the problem.

The following conclusions can be stated:

  • 1.

    Divergent mode of stability loss for a membrane is not symmetric.

  • 2.

    Experiments show

References (9)

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