A study on wrinkling characteristics and dynamic mechanical behavior of membrane

Acta Mech. Sin. (212 28(1:21 21 DOI 1.17/s149-11-512-2 RESEARCH PAPER A study on wrinkling characteristics and dynamic mechanical behavior of membrane Yun-Liang Li Ming-Yu Lu Hui-Feng Tan Yi-Qiu Tan Received: 8 June 21 / Revised: 2 April 211 / Accepted: 7 June 211 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 212 Abstract An eigenvalue method considering the membrane vibration of wrinkling out-of-plane deformation is introduced, and the stress distributing rule in membrane wrinkled area is analyzed. A dynamic analytical model of rectangular shear wrinkled membrane and its numerical analysis approach are also developed. Results indicate that the stress in wrinkled area is not uniform, i.e. it is larger in wrinkling wave peaks along wrinkles and two ends of wrinkle in vertical direction. Vibration modes of wrinkled membrane are strongly correlated with the wrinkling configurations. The rigidity is larger due to the heavier stress in the part of wrinkling wave peaks. Therefore, wave peaks are always located at the node lines of vibration mode. The vibration frequency obviously increases with the vibration of wave peaks. Y.-L. Li ( School of Transportation Science and Engineering, Harbin Institute of Technology. Post-doctoral Research Center in Civil Engineering, Harbin Institute of Technology, 159 Harbin, China e-mail: Liyl-hit@163.com M.-Y. Lu Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China H.-F. Tan Center for Composite Materials and Structure, Harbin Institute of Technology, 151 Harbin, China Y.-Q. Tan School of Transportation Science and Engineering, Harbin Institute of Technology, 159 Harbin, China Keywords Space membrane structure Wrinkle Vibration 1 Introduction The membrane wrinkle is a local buckling phenomenon formed under stress and boundary conditions, and it exists commonly in membrane structures, such as space inflatable structure, tension membrane structure, small sandwich membrane structure, human skin, wizened fruit and microcosmic cell wall. To evaluate and predict shapes and stabilities of membrane structures, the membrane with wrinkle should be considered. There are two aspects in wrinkling study: one focuses on the stress distribution after the formation of wrinkle rather than its specific pattern of deformation, for example, study on the wrinkle of the tension membrane structures. The other one focuses on the specific wrinkling pattern (wavelength and amplitude of wrinkle, for example, study on the wrinkle of high-precision inflatable deployment membrane structures. With the development of high-accuracy inflatable deployment structures, the wrinkling specific patterns becomes the main research direction in the field of membrane wrinkles [1, 2]. At present, there are two major methods on the analytical predictions of the membrane wrinkles: tension field theory and buckling analysis. The tension field theory was proposed by Wagner [3] in 1929. In this theory, the bending rigidity of membrane was ignored, and it was held that the wrinkling direction was along the main principal stress and the minor principal stress in the vertical wrinkle direction was zero. Many wrinkling analytical models were established based on this theory [4 11]. The basic idea of the tension field theory was to modify the constitutive equations with the assumption of zero minor principal stress vertical to the wrinkle direction, so as to avoid occurrence of the compression stress in the membrane. The right stress distribu-

22 Y.-L. Li, et al. tion and directions in the wrinkled area could be obtained by numerical solution. Its major drawbacks are that the information of the wavelength and amplitude of the membrane wrinkle could not be gained, and the numerical solution would be difficult when the revision is excessively complex. In the buckling analytical method, the slight bending rigidity of membrane is considered so that the specific information of the membrane wrinkle can be got. Cerda and Mahadevan [12] obtained the quantitative relationship of the wavelength and amplitude by the balance of the bending and tension strain energy in the wrinkled area, and they analyzed the wrinkles of fruit and human skins. Wesley and Sergio [13,14] considered the stress balance relationship in wrinkled area and the action of minute critical compression stress in the vertical wrinkle direction, and they deduced the formula of the wrinkle wavelength and amplitude, and made verification by numerical simulation and experiments. Primary studies have been done on the dynamic characteristics of wrinkled membrane. Kukathasan [15] performed numerical simulation for the self-vibration characteristics of wrinkled membrane, in the iterative membrane properties (IMP method, and compared the vibration effect in different wrinkling modes. In addition, he analyzed the experimental results in consideration of air damping. Hossai and Jenkins [16,17] studied the vibration characteristics of wrinkled triangular and annular membrane by comparison of experimental results. In their studies, penalty parameter modified material (PPMM was used in the simulation. Penalty parameter was a small rigidity value close to zero in the direction of small main-stress in a plane, so as to eliminate the singularity of rigidity matrix for calculation convergence. They made a wrinkling prediction by PPMM. It could be seen through dynamic analysis that there was a great difference between the vibration modes of membranes with wrinkles and without wrinkles, and both inherent frequencies were different. In analyses above on vibration characteristics of wrinkled membrane, only the effect of stress changes in wrinkling area was considered on membrane dynamic characteristics, while the effect of out-of-plane deformation was ignored on the dynamic characteristics of membrane after wrinkling formation. In this paper, an eigenvalue equation of wrinkled membrane vibration was established firstly in consideration of effect of wrinkling out-of-plane deformation, and through stress distribution law in membrane wrinkled area, an analytical model of shear membrane wrinkles was obtained. According to the model, the numerical flow of dynamic analysis for the wrinkling out-of-plane deformation was proposed. The correlation of wrinkle configuration and virbration modes, as well as the effect of wrinkling out-ofplane deformation were analyzed in combination with regularities of stress distribution in the wrinkling area. 2 Vibrating differential equation of wrinkled membrane The large out-of-plane deformation occurs after wrinkle formation for membrane. The vibration of the wrinkled membrane vibrates by the wrinkle configuration as the equilibrium position. The vibrating differential equation of wrinkled membrane based on the Lagrange equation was established. In the process of vibration, the bending deformation potential of membrane during vibration is [18] U 1 = D 2 U 2 = { ( 2 w 2 2(1 v [( 2 w 2 2 w 2 ( 2 w 2 ]} dxdy. (1 And the strain potential energy is given by [18] Et ( ε 2 2(1 v 2 x + ε 2 y + 2vε x ε y + 1 v 2 γ2 xy dxdy. (2 If out-of-plane rather than inner-plane vibration of membrane is considered only, the expression of strain should be [19] ε x = 1 ( w 2 1 w 2, 2 2( ε y = 1 ( w 2 1 w 2, (3 2 2( γ xy = w w w w. where w represents vibration configuration function and w is the expression of the wrinkling out-of-plane deformation, both of which are functions of coordinates (x, y. The membrane thickness is t and membrane planar density is ρ. Substituting Eq. (3 into Eq. (2, strain potential energy of membrane is given by Et {[( w 2 ( w 2 ] 2 U 2 = 8(1 v 2 [( w 2 ( w 2 ] 2 + [( w w 2 ( w w 2 ] 2v + ( w w 2 w w +2 + 2( 2 4(1 v w w w w } dxdy. (4 The expression of kinetic energy of membrane vibration is T = 1 [ (w w ] 2 2 ρt dxdy t = 1 ( w 2 2 ρt dxdy. (5 t

A study on wrinkling characteristics and dynamic mechanical behavior of membrane 23 Assuming vibration and wrinkling configurations are presented in the following progression style w(x, y, t = w i (x, yq i (t, w (x, y = w i (x, y, n n and substituting Eq. (6 into Eqs. (1, (4 and (5, the potential energy expression gained in the process of vibration is as follows V = U 1 + U 2 = 1 2 (k i jklq i jkl + k i j q i j + k i jkl, (7 and the kinetic energy expression is T = 1 2 m i j q i q j. (8 In Eq. (7, the expressions of k i jkl, k i j, k i jkl, m i j are as follows Et ( wi w j w k w l k i jkl = 4(1 v 2 + w i w j w j w k w k w l w l +2 w i [ 2 w i 2 w j k i j = D 2 + 2 w i 2 w j 2 2 2 (6 dxdy, (9 +2v 2 w i 2 w j 2 + 2(1 v 2 w i 2 w ] j dxdy 2 Et [ wi w j w w i j + 2(1 v 2 + w i + w i w j w j w i w i w ( j + v wi w j w w i j w j w i w j +2(1 v w i k i jkl = Et ( w i w j w k w l 4(1 v 2 + w i +2 w i m i j = ρt w j w j w k w k w l w ] j dxdy, (1 w l dxdy, (11 w i w j dxdy. (12 According to the Lagrange equation, L d L =, (13 q i dt q i where L = T V. (14 By substituting Eqs. (7 and (8 into Eq. (14 and combining Eq. (13, the following can be got m i j q i + k i j q i + k i jkl q i q k q l =. (15 The differential equation of linear vibration could be obtained regardless of the nonlinear part of Eq. (15, then m i j q i + k i j q i =. (16 Assuming q i =a i sin ωt and substituting it into Eq. (16, the following can be got (k i j ω 2 m i j a i =. (17 Its matrix form can be obtained as (K ω 2 MG =, (18 where stiffness matrix K and mass matrix M are represented in Eqs. (1 and (12, respectively. Vector G is the vibrating eigenvector, i.e. vibrating mode. According to Eq. (1, the stiffness matrix is combined with the effect of wrinkle outof-plane deformation in the membrane equilibrium position. The natural frequency and modes of wrinkled membrane vibration can be obtained by solving the eigenvalue and eigenvector of Eq. (18. The eigenvalue problem can be solved based on subspace iterative method or Lanczos method. K = K e, (19 where element stiffness matrix K e is constituted of three parts [19] K e = K el + K eσ + K enl. (2 And K el = B T L DB LdS, K eσ = B T NL σds, K enl = B T L DB NLdS, (21 where K el, K eσ, K enl, σ, B and D stand for linear stiffness matrix, stiffness matrix of initial stress, large displacement stiffness matrix, initial stress, strain matrix and elastic matrix, respectively. It is seen from expression (1 for the element stiffness matrix of wrinkled membrane, that dynamic analysis of wrinkled membrane is conducted by two steps: first, the membrane equilibrium configuration is acquired under effect of external load, i.e. wrinkling configuration; second, the component of stiffness matrix with wrinkle deformation effect can be obtained based on renewal of geometric model by wrinkle deformation, and finally, natural frequency and modes of system vibration could be obtained by the eigenvalue analysis.

24 Y.-L. Li, et al. 3 Stress distribution of wrinkled membrane Generally take the wrinkle element of half-wavelength to study. h stands for the wrinkle element length, and λ is its width of half-wavelength of wrinkle, E is Young s modulus, t is thickness, and ν is Poisson s ratio. Its boundary condition is the freely-supporting boundary along wrinkles while the wrinkling direction perpendicular to wrinkle is the fixing boundary. The boundary condition is w =, x =, x = λ, w =, y =, y = h. (22 Mean value of normal stress along wrinkles is represented σ x = 1 h σȳ = 1 λ h λ σ x dy, σ y dx. (23 Assuming the out-of-plane deformation function under applied load is w(x, y = A sin πx λ sin πy h, (24 which satisfies all the boundary conditions. In order to simplify the problem, assuming the deformation form of initial membrane flaw is equal to that of wrinkle, i.e. the form of initial flaw is as follows w (x, y = A sin πx λ sin πy h, (25 where A is the membrane out-of-plane deformation amplitude induced by initial flaw. Membrane compatible equations with initial deformation is given by [2] 1 E 4 F = ( 2 w 2 w 2 2 + 2 2 w 2 w 2 w 2 w 2 2 2 w 2 w 2 w 2 2. (26 2 In combination with equations of out-of-plane deformation equation (24 and initial flaw equation (25, items in Formula (26 are 2 w = Aπ2 πx πy cos cos hλ λ h, 2 w 2 2 w 2 = Aπ2 λ 2 = Aπ2 h 2 πx πy sin sin λ h, πx πy sin sin λ h, (27 2 w = A π 2 hλ cos πx πy cos λ h, 2 w = A π 2 sin πx πy sin 2 λ 2 λ h, 2 w = A π 2 sin πx πy sin 2 h 2 λ h. Substituting Eq. (27 into Eq. (26 gives 1 E 4 F = π4 A ( 2h 2 λ (A + 2A cos 2πx 2πy + cos, (28 2 λ h where F is the stress function of wrinkled membrane. Assuming the particular solution to Eq. (28 is F 1 = k 1 cos 2πx λ + k 2 cos 2πy h, (29 and by substituting Eq. (29 into Eq. (28, clearing up and comparing coefficients, then k 1 = EA(A + 2A λ 2 32h 2, k 2 = EA(A + 2A h 2 32λ 2. Then substituting it into Eq. (29 gives ( λ 2 2πx cos h2 λ F 1 = EA(A + 2A 32 + h2 2πy cos λ2 h (3. (31 Before wrinkle formation of membrane, homogeneous solution to Eq. (28 is corresponding to the stress state i.e. the form of Eq. (23, then, the homogeneous solution is y 2 F 2 = σ x 2 + σ x 2 ȳ 2, (32 where the general solution to Eq. (28 could be obtained y 2 F = σ x 2 + σ ȳ ( λ 2 2πx cos h2 λ x 2 2 + EA(A + 2A 32 + h2 2πy cos λ2 h. (33 And the stress in membrane wrinkled zone can be determined by σ x = 2 F 2, σ y = 2 F 2, τ xy = 2 F. (34 Now solving A in Eq. (33 by the Galerkin Principle, the Galerkin equation is given by h λ Q(x, y sin πx λ And the expression of Q(x, y is Q(x, y = D 4 w t [ σ x 2 (w + w 2 πy sin dxdy =. (35 h 2 (w + w 2 (w + w +σ y + 2τ 2 xy ]. (36 Substituting Eqs. (24, (25 and (34 into Eqs. (36 and (35 for integration and simplifying, the standard form of simple cubic equation about A can gained as follows A 3 = aa 2 + ba + c, (37 where expressions of parameters a, b and c are

A study on wrinkling characteristics and dynamic mechanical behavior of membrane 25 a = 3A, b = 2A 2 16tλ2 h 2 (h 2 σ x + λ 2 σȳ + 16π 2 D(h 2 + λ 2 2, Eπ 2 t (h 4 + λ 4 c = 16A λ 2 h 2 (h 2 σ x + λ 2 σȳ. Eπ 2 (h 4 + λ 4 Solution to Eq. (37 is given by (38 Fig. 1 Stress distribution of the wrinkled area A = 3 X1 + 3 X 2 + a 3, (39 where expressions of X 1 and X 2 are X 1,2 = 1 [ 2a 3 2 27 + ab 3 + c (2a 3 ± 27 + ab 2 ( 3b + a 2 3 ] 3 + c 4. (4 9 The stress expression in wrinkled area with initial flaw can be obtained by Eqs. (33 and (34 σ x = σ x EA2 π 2 (1 + 2γ 8λ 2 σ y = σȳ EA2 π 2 (1 + 2γ 8h 2 τ xy =, cos 2πy h, cos 2πx λ, (41 where A is represented by Eq. (39, and γ = A /A and it is flaw coefficient. Hence, the maximum and minimum of the compressive stress perpendicular to wrinkles and tensile stress along wrinkles can be obtained as σ x max = σ x EA2 π 2 (1 + 2γ 8λ 2, σ x min = σ x + EA2 π 2 (1 + 2γ 8λ 2, σ y max = σȳ + EA2 π 2 (1 + 2γ 8h 2, (42 σ y min = σȳ EA2 π 2 (1 + 2γ. 8h 2 If making γ =, stress expression of membrane wrinkled area without initial flaw can be determined by Eq. (41 σ x = σ x EA2 π 2 8λ 2 σ y = σȳ EA2 π 2 8h 2 τ xy =. cos 2πy h, cos 2πx λ, (43 The wrinkling stress distribution along x-, y-directions by Eq. (43 is illustrated in Fig. 1, which shows that the tensile stress is maximal in the location of wrinkle wave crest along x-direction, the larger compressive stress occurs in both wrinkle ends, and minimal stress occurs near center along y-direction. 4 Numerical analysis on membrane wrinkles 4.1 Analytical model of shear membrane wrinkles An analytical model of the shear membrane wrinkle is established as shown in Fig. 2. The length and width of the membrane are L and h, respectively. The material parameters are: Young s modulus E, thickness t and Poisson s ratio ν, with the boundary conditions of fixing the upper and lower ends and free left and right ends. The extending distance up to the upper end is δ 2 and the shear distance from upper end along the x-axis is δ 1. Then, an inclined wrinkle occurs by an included angle θ with hemline. The boundary conditions are given by u = δ 1, v = δ 2, x L, y = h, u =, v =, x L, y =. Fig. 2 A rectangular membrane under shear 4.2 Flow of numerical analysis on membrane wrinkles (44 The membrane configuration changes after wrinkle formation, and the great out-of-plane deformation of membrane equation of wrinkled membrane and the stress distribution of membrane occur relative to the initial plane configuration. Hence, the dynamic analysis should include the change of membrane configuration and its induced changes of mass matrix and stiffness matrix, that is, the dynamic analysis should be completed based on the wrinkle configuration. The flowchart for the numeric analysis including the dynamic analysis of wrinkled membrane is shown in Fig. 3.

26 Y.-L. Li, et al. Fig. 3 Flowchart of dynamic analysis for wrinkled the membrane The flowchart consists of two main procedures: (1 Static analysis: Acquisition of membrane wrinkle configuration and corresponding stress distribution. The analytical process of membrane wrinkles is established based on ANSYS software. Shell 63 element with four nodes and six degrees of freedom at each node is adopted, and is used to analyze large deformation with effect of the stress rigidity. By applying the initial tension distance δ 2 along y-direction, the membrane is made with the definite initial rigidity in convenient for analysis on convergence. In order to trigger the formation of wrinkles for the post-buckling analysis, it is crucial to apply the initial imperfection. The direct turbulence method is adopted in this paper to apply the initial flaw, that is, the distributed outof-plane stresses vertical to membrane are applied directly on the nodes, and numbers of the plus-minusvalues of the out-of-plane stresses are equal, so as to ensure the composite force of distributed turbulence force is zero. The outof-plane deformation of membrane under turbulence forces and its thickness are in the same magnitude. Then the initial non-linear analysis is performed applying a small shear displacement δ 1 along x-direction (δ 1 1.6 δ 2 is rational actually. The initial turbulence forces are removed to eliminate the effect of initial imperfection on the post buckling analysis. Then the shear displacement δ 1 (δ 1 = δ 1 δ 1 is applied to start nonlinear post buckling analysis and to get the membrane wrinkle confifuration without initial imperfection. The calculation can be proceeded with the post buckling process by the displacement controlled method and the Newton Raphson iterative method. The analytical convergence can be improved by adjusting the number of load substeps in combination with dichotomy and restart of analysis, as well as the initial shear displacement δ 1. (2 Dynamic analysis: Acquisition of the natural frequency and vibration modes of wrinkled membrane. The impact of wrinkle configuration on dynamic characteristics is considered by model renewal, and the impact of stress in wrinkled membrane on dynamic characteristics is considered by the initial stress effect. 5 Experimental verification The shear membrane model illustrated in Fig. 2 was used with membrane size 38 mm 12 mm, Young s modulus E = 3.53 GPa, Poisson s ratio ν =.34, density ρ = 1 4 kg/m 3 and thickness t = 5 µm. The photogrammetric procedure was used for measurement of the wrinkle deformation. The experimental equipment, shown in Fig. 4, was made up of supporting frame, measuring scale and adjusting bolt. Canon EOS 35D camera and PhotoModeler Pro5 analytical processing software were used in the experiment. This programme could handle photos taken from different angles and get the three-dimensional coordinates of targets. The measuring scale provided a standard reference length and coordinates. Different tension stresses and shear deformation could be got by adjusting bolts. The number of targets was 3 76. The wrinkling deformation modes corresponding to shear displacements of 1 mm, 2 mm, 3 mm from the experiment are shown in Fig. 5. It can be seen from the figure that the half-wave number of wrinkles is 24 in different shear displacements. The out-of-plane deformation curves of longitudinal mid-line from simulation and experiment are shown in Fig. 6 when the shear displacement is 1 mm. It can be seen that the results of numerical analysis and experiment ideally match each other, which proves the rationality of numerical analysis method established.

A study on wrinkling characteristics and dynamic mechanical behavior of membrane 27 6 Dynamic characteristics of shear wrinkled membrane 6.1 Dynamic analysis of membrane without wrinkles Fig. 4 Experimental study of wrinkles by photogrammetric procedure For comparison, dynamic characteristics of one-way tensile membrane without wrinkles are firstly analysed by stretching the upper end of the membrane by.1 mm. By then, there is no wrinkling deformation occurring. Modes and natural frequencies of the first eight orders are shown in Fig. 7. Fig. 7 Vibration modes and natural frequencies of rectangular membrane Fig. 5 Wrinkling configuration under different shear displacements (experimental result. a δ1 = 1 mm, δ2 =.1 mm; b δ1 = 2 mm, δ2 =.1 mm; c δ1 = 3 mm, δ2 =.1 mm The dashed lines in the figure indicate the locations of node lines. Because the right and left sides of the membrane are free ones, the first and second-order modes of the membrane are vibrated locally near two free sides, and the integral vibration occurs from the third-order mode. The number of node lines increases successively and the lines are parallel to boundary. 6.2 Relationship of vibration modes and wrinkle configurations Fig. 6 Comparison of wrinkle amplitudes (δ1 = 1 mm, δ2 =.1 mm The modes of the first four orders, shown in Fig. 8, are gained by analysis of membrane dynamic characteristics under the conditions of.1 mm initial tensile distance and.5 mm shear distance. Lines jointed with boundaries are node lines. It is obvious from the vibration-mode figure that the vibration modes are totally different from that of nonwrinkling membrane shown in Fig. 7.

28 Fig. 8 Vibration modes of wrinkled membrane. a 1st vibration mode; b 2nd vibration mode; c 3rd vibration mode; d 4th vibration mode Figure 9 shows the comparison between vibration mode of longitudinal center line of wrinkled membrane and wrin- Y.-L. Li, et al. kle configuration of longitudinal center line at tensile displacement of.1 mm and shear displacement of.5 mm. The location of node lines is the same as that of arrowheads. The number of node lines decreases with increase of modes. Relevance between vibration modes and wrinkle configurations can be seen clearly from the comparison of longitudinal center line mode and wrinkle configurations. From Fig. 9, symmetric modes and antisymmetric modes occur alternately, and the location of node lines is corresponding to that of the wave peak of wrinkle configuration (exclusive of the node lines on symmetry axis of antisymmetric mode. There are three types of vibration in vibration modes by node lines: I-type local vibration, which vibrates equidirectionally in the range of 1/2 wrinkle wavelength with node lines locating at wrinkle wave peaks; II-type local vibration, which vibrates equidirectionally in the range of 3/4 wrinkle wavelength with node lines locating at wrinkle wave peaks and middle facial lines; and III-type local vibration, which vibrates equidirectionally in the range of one wrinkle wavelength with node lines locating at wrinkle wave peaks. The rest may be deduced by analogy that homodromous vibration will occur in the higher-order modes with 3/2 wrinkle wavelength, 7/4 wrinkle wavelength, 2 wrinkle wavelength, and so on. The numbers of three different local vibrating modes of first eight modes are shown in Table 1. From the number of three local vibration modes, vibration ways are more complicated for the high-order modes, and more wave peaks involve in vibration. Fig. 9 Comparison of membrane vibration modes and wrinkle configurations. a 1st vibration mode (symmetry; b 2nd vibration mode (antisymmetry; c 3rd vibration mode (symmetry; d 4th vibration mode (symmetry The wave peak means the largest out-of-plane deformation in the wrinkle range and larger tensile stress according to Eq. (43 and Fig. 1. It is known according to Eq. (21 that a superior local stiffness in wrinkle wave peak is formed. As a result, the wrinkle wave peak is mostly located on node lines during membrane vibration, and no vibration occurs. Un- der the higher-order mode, the corresponding vibration frequency increases with the vibration of wrinkle wave peaks involving. It can be seen by this that the out-of-plane deformation of wrinkles and the regional stress distribution both determine vibration mode and wrinkle configuration of the wrinkled membrane.

A study on wrinkling characteristics and dynamic mechanical behavior of membrane 29 Table 1 Distribution of different local vibrations Modes Local modes I II III f1 23 f2 2 2 f3 17 4 f4 18 4 f5 19 2 f6 16 2 2 f7 15 4 f8 12 2 4 6.3 Effects of wrinkle amplitudes on vibration frequencies Variation between the first fourth orders of natural frequencies and shear distances with increase of shear deformation at the initial tensile displacement of.1 mm is shown in Fig. 1 (scales of upper abscissa axis are ratio of maximal outof-plane wrinkle deformation according to shear distances to membrane thickness. Point P in the figure represents the critical state of membrane full of wrinkles with the lowest basic frequency. Then with the shear displacement and oscillating variation, its frequency increases. All orders of frequencies apparently decrease compared with the circumstances without shear deformation. Vibration modes according to P1 P4 are shown in Fig. 11, where only locations of node lines are shown. There is no change in number of node lines other than local vibration on boundaries under large shear displacement. Fig. 1 Shear displacements and natural frequencies 7 Conclusions The dynamic analytical equations for wrinkled membrane are established based on the Lagrange equation, and the effect of the out-of-plane deformation and distribution of Fig. 11 1st mode under different shear displacements. a Mode node lines of P1; b Mode node lines of P2; c Mode node lines of P3; d Mode node lines of P4 stresses on dynamic characteristic of wrinkled membrane are considered. The stress distribution in the membrane wrinkling area is obtained by the theoretical analysis. The stress is larger in wrinkle wave peak along wrinkle direction, and larger in two ends of the wrinkle in vertical direction. Numerical approach and mechanical model of dynamic analysis on shear membrane wrinkles have been developed, including two different processes, i.e. static and dynamic analyses. Wrinkle configurations and stress distribution can be obtained by static analysis and dynamic characteristics after wrinkle formation can be obtained by dynamic analysis. Photogrammetry is introduced into the experiments so as to study wrinkle configurations of shear membrane. The experimental result agrees well with simulation result, which verifies rationality of simulation analysis proposed in the paper. Results show that vibration modes of wrinkled membrane are strongly correlated with wrinkle configurations. The local stiffness of wrinkle wave peaks is larger because of large stress at the wrinkling wave peak. Therefore, the wave peaks are mostly located at node lines. Vibration frequency apparently increases with wave peak vibration involving. With the wrinkle out-of-plane deformation, the natural frequency of vibration changes in oscillation. References 1 Stein, M., Hedgepeth, J.M.: Analysis of partly wrinkled membranes. NASA, Tech. Notes, D-813 (1961

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