# Theoretical and Numerical Studies on Damped Nonlinear Vibration of Orthotropic Saddle Membrane Structures Excited by Hailstone Impact Load

Authors: Changjiang Liu; Fan Wang; Jian Liu; Xiaowei Deng; Zuoliang Zhang; Haibing Xie
Journal: Shock and Vibration
 (2019)

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## Abstract

The orthotropic membrane structures have been popular in architectural structures. However, because of its lightweight and small stiffness, large nonlinear deflection vibration may occur under impact load, which leads to structural failure. In this paper, the governing equations of the large deflection nonlinear damped vibration of orthotropic saddle membrane structures excited by hailstone impact load are proposed according to the von Kármán’s large deflection theory and solved by applying the Bubnov–Galerkin method and the method of KBM perturbation. The approximate theoretical solution of the frequency function and displacement function of the large deflection nonlinear damped vibration of saddle membrane structures with four edges fixed excited by hailstone impact was obtained. The analytical examples proved that the mode shape function (equation (43)) can be applied to calculate the single-order mode shapes and the total superposed mode shapes of the damped large nonlinear deflection vibration of orthotropic saddle membrane structures excited by hailstone impact load succinctly. In addition, we compare and analyze the results of vibration frequency, amplitude, time histories, and total displacement of membrane structures with different pretensions and arch-to-span ratios under the impact of differently sized hailstones. The correctness of the analytical theory is verified by comparing with the results of numerical simulation. According to the results of this paper, we put forward some suggestions for the vibration control and dynamic design of practical spatial membrane structures.


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## Body

## 1. Introduction

The membrane structure which is made of orthotropic membrane materials has been popular in architectural structures, and the saddle membrane structure is one of the most common shapes of the double-curved membrane [1, 2]. However, because of its lightweight, large flexibility, and small stiffness, it is very sensitive to impact load. Large nonlinear deflection vibration occurs under impact load, which may lead to structural failure [3, 4]. Thus, it is necessary to study damped large nonlinear deflection vibration of orthotropic saddle membrane structures excited by hailstone impact load.In recent decades, more and more attention has been focused on the dynamic characteristics of the membrane. By applying the Hamilton principle and Galerkin method, Shin et al. [5, 6] obtained the natural frequencies and mode shapes of the free vibration for an axially moving membrane. The results showed that the translating speed, aspect ratio, and boundary conditions have significant effects on the in-plane vibrations of the moving membrane. Pan and Gu [7] adopted D’Alembert’s principle to deduce the free oscillating system’s equivalent fundamental frequency of the square tensioned membrane. The effects of prestrain, size, elastic ratio, density, relative amplitude, and dead load on the nonlinearity of square pretensioned membrane were studied. Zheng et al. [8], Liu et al. [9], and Li et al. [10] investigated the large deflection nonlinear free vibration of orthotropic rectangular membrane structure by applying von Kármán’s large amplitude theory, D’Alembert’s principle, Bubnov–Galerkin approximate method, and Lindstedt–Poincaré perturbation method and obtained the approximate analytical solution in power series of the nonlinear vibration frequency function and the displacement function of the rectangular membrane with four edges fixed. However, they only studied the free vibration of planar rectangular membrane structures.On the basis of researches of the nonlinear free vibration of membrane structures, the investigations of the nonlinear forced vibration of membrane structures were performed. Gonçalves et al. [11] derived the equations of motion of the prestretched hyperelastic isotropic membrane with finite deformations and lateral pressure and obtained analytically the functions of natural frequencies and mode shapes of the membrane. The results show the stretching ratio significantly affected the linear and nonlinear vibrations of the membrane. Zheng et al. [12] applied D’Alembert’s principle and the momentum theorem to derive the fundamental equations of forced vibrations of orthotropic membranes, which were solved according to the Lindstedt–Poincaré perturbation method, and obtained the formula of impact load and nonlinear forced vibration deflection of rectangular membranes with four edges fixed. However, the damping was not considered. By applying the Krylov–Bogolubov–Mitropolsky (KBM) perturbation method to solve the governing equations of large-amplitude nonlinear vibration of rectangular orthotropic membranes with viscous damping, Liu et al. [13] obtained the asymptotic analytical solutions for the frequency and displacement function of the planar rectangular membrane structure. However, the external load was not taken into consideration. According to the Föppl large deflection theory, Galerkin method, and multiple scale perturbation method, Zheng et al. [14] investigated the dynamic response of rectangular prestressed membrane subjected to concentrated impact load. Based on the von Kármán large deflection theory, D’Alembert’s principle, the Bubnov–Galerkin method, and perturbation method, Liu et al. [15–17] studied nonlinear forced vibration of pretension rectangular orthotropic membrane structures with damping and without damping under single impact load. Based on the stochastic pulse theory and the random vibration theory, Zheng et al. [18] and Li et al. [19, 20] investigated the stochastic vibration problem of the orthotropic membrane subjected to random impact load through experimental and theoretical researches. Based on thin-plate theory and the Galerkin method, Li et al. [21] investigated the dynamic response of rectangular prestressed membrane subjected to uniform impact load theoretically and experimentally. The aforementioned studies, however, are limited to planar rectangular membrane structures.On the current status, the studies on saddle membrane structure are concentrated on wind or rain loads. Yang and Liu [22] and Li and Sun [23] studied the aerodynamic critical unstable wind velocity of saddle membrane structure by applying the nonmoment theory and the potential flow theory. The aerodynamic interaction equations of the membrane structure were obtained and simplified by applying the Bubnov–Galerkin approximate method. By studying the free vibrations of membrane structures with the static and the dynamic effects of wind and snow, Lazzari et al. [24] analyzed the structural failure mechanisms of the roof of the Montreal Stadium membrane structure. Rizzo et al. [25] and Rizzo and Sepe [26] measured the pressure of the incoming wind on hyperbolic paraboloid roofs by conducting tests and finite element analyses. The possibility of defining equivalent static pressure fields able to reproduce the envelope of dynamic displacements of the cables net was explored. Wu et al. [27] studied the aeroelastic instability mechanism of a tensioned membrane structure. The response and wind velocities above two closed-type saddle-shaped tensioned membrane structures, with the different pretension levels, were measured in uniform flow and analyzed. The results indicate that the aeroelastic instability is caused by vortex-induced resonance. Xu et al. [28, 29] and Liu et al. [30] studied nonlinear wind-induced aerodynamic stability of orthotropic saddle membrane structures by establishing the interaction governing equations of wind-structure coupling based on von Kármán’s large amplitude theory and D’Alembert’s principle. They determined the critical velocity of divergence instability, by judging the stability of the characteristic equation of the system. Cui et al. [31] applied the Eulerian–Eulerian model according to multiphase flow theory, and the saddle membrane structure response analysis under the simultaneous actions of wind and rain was conducted. The influences of changes in wind speed and rain intensity on the saddle-shaped membrane structure response were compared. There are no researches about the problem of saddle orthotropic membranes under impact load.In this paper, the approximate formulas of hailstone terminal velocity were substituted into the governing equations of the large deflection nonlinear damped vibration of orthotropic saddle membrane structures excited by impact load. And, solving the governing equations by applying the Bubnov–Galerkin method and the method of KBM perturbation, the approximate theoretical solution of the frequency function and displacement function of the large deflection nonlinear damped vibration of saddle membrane structures with four edges fixed excited by hailstone impact was obtained. In analytical examples, the dynamic responses of saddle membrane structures with different pretension levels and arch-to-span ratios excited by the impact of different diametral hailstones were compared and analyzed separately. The correctness of the analytical theory is verified by comparing with the results of numerical simulation. In addition, the results of this paper can be applied in computation for the vibration control and dynamic design of practical spatial membrane structure under impact load.


## 2. Modeling of Saddle Membrane Structure

In this paper, we study a saddle, namely, hyperbolic paraboloid, membrane structure with four edges simply supported under an impact load. The theoretical model of saddle membrane structure is shown in Figure1. The orthogonal axes x and y are the two different Young’s modulus fiber directions of orthotropic saddle membrane structures. a and b, respectively, are the spans in x and y axes. N0x and N0y, respectively, denote the pretension in x and y axes. f1 and f2, respectively, are the midspan arch in x and y axes. The point O is the center of the plane xoy. The sphere H is a hailstone; v0 denotes the velocity of the hailstone; (x0, y0) is the impact point on membrane surface.Figure 1
Pretensioned orthotropic saddle membrane model under an impact load with four edges simply supported.The saddle membrane model can be represented by [29](1)z0x,y=f2x−a/22a/22−f1y−b/22b/22,where z0 denotes the initial surface function of saddle membrane structure.According to equation (1), the two initial principal curvatures in x and y directions are(2)k0x=∂2z0∂x2=8f2a2,k0y=∂2z0∂y2=−8f1b2.With the action of the pretensionsN0x and N0y [28], we can obtain(3)k0xN0x+k0yN0y=0.


## 3. Dynamic Governing Equations

According to the von Kármán’s large deflection theory and D’Alembert’s principle, the compatible equation and dynamic motion equation of orthotropic saddle membrane structures are [28](4)ρ0∂2w∂t2+c∂w∂t−Nx+N0xkx−Ny+N0yky+2Nxykxy=px,y,t1E1h∂2Nx∂y2+1E2h∂2Ny∂x2−μ1E1h∂2Nx∂x2−μ2E2h∂2Ny∂y2−1Gh∂2Nxy∂x∂y,=∂2w∂x∂y2−∂2w∂x2∂2w∂y2−k0x∂2w∂y2+k0y∂2w∂x2,where ρ0 is the areal density of membrane; Nx and Ny, respectively, are the stress increments in x and y directions; Nxy is the shear stress; N0x and N0y, respectively, are the initial stress in x and y directions; w=wx,y,t is the lateral deflection; h is the thickness of membrane; E1 and E2, respectively, are Young’s modulus in x and y directions; G is the shear modulus; μ1 and μ2, respectively, are Poisson’s ratio in x and y directions; kx and ky, respectively, are the two principal curvatures in x and y directions; k0x and k0y, respectively, are the two initial principal curvatures in x and y directions; c is the damping coefficient of structure; px,y,t is the impact load in z direction.According to the basic theory of plates and shells, the principal curvatures inx and y directions are [21](5)kx=k0x+Δkx=k0x+∂2w∂x2,ky=k0y+Δky=k0y+∂2w∂y2,where Δkx and Δky denote the principal curvature increments in x and y directions, respectively.By means of Airy’s stress functionφ=φx,y,t, we can obtain [30](6)Nx=h∂2φ∂y2,Ny=h∂2φ∂x2,Nxy=−h∂2φ∂x∂y.The effect of the shear stress is so small that we may assume thatNxy = 0. Therefore, according to equation (6), we can obtain(7)∂2Nx∂x2=∂2Ny∂y2=h∂4φ∂x2∂y2=−∂2Nxy∂x∂y=0.By substituting equation (3) and equations (5)–(7) into equation (4), we can obtain(8)ρ0∂2w∂t2+c∂w∂t−h∂2φ∂y2+N0x∂2w∂x2−k0xh∂2φ∂y2−h∂2φ∂x2+N0y∂2w∂y2−k0yh∂2φ∂x2=px,y,t,(9)1E1∂4φ∂y4+1E2∂4φ∂x4=∂2w∂x∂y2−∂2w∂x2∂2w∂y2−k0x∂2w∂y2−k0y∂2w∂x2.Assume that the membrane does not bear an external load apart from the impact load; then, when the point of impact lies in the coordinates on (x0,y0) (as shown in Figure 1), the equation becomes [10](10)px,y,t=Ftδx−x0y−y0,where Ft is the impact force and δx,y is the impulse function.Assume that the initial displacement of the membrane is zero before the membrane is impacted by hailstone; when the hailstone is just in contact with the membrane, at the moment oft = 0, they have the same velocity, and this velocity is the initial velocity of the membrane, so the initial conditions are(11)wx0,y0,tt=0=0,∂wx0,y0,t∂tt=0=v0.Impact loads are short-duration loads, so the maximum displacement amplitudewmax depends principally upon the magnitude of the applied impulse I=∫0tFτdτ and is not strongly influenced by the form of the load; thus, we can obtain the relational expression between Ft and wx,y,t by the impulse theorem:(12)∫0tFτdτ=Mv0−M∂wx0,y0,t∂t,where M is the mass of hailstone.Derivation calculus to equation (12) yields(13)Ft=−M∂w2x0,y0,t∂t2.Equations (8)–(13) are the fundamental equations applied in the analysis of saddle membrane excited by impact load.According to the approximate formulas of hailstone terminal velocity [32], we can obtain the mass and velocity of hailstones (as shown in Table 1 and Figure 2).Table 1
Mass and velocity of hailstone with different diameters.
Diameter (cm)
1.0
2.0
3.0
4.0
5.0
6.0
Mass (kg)
0.0005
0.0038
0.0127
0.0302
0.0589
0.1017
Mass difference
0.0033
0.0089
0.0175
0.0287
0.0428
Velocity (m/s)
16.12
21.38
25.22
28.36
31.06
35.74
Velocity squared difference
197.25
178.94
168.24
160.43
312.62Figure 2
Mass and velocity of hailstone in different diameters.The boundary conditions of the saddle membrane structure with four simply supported edges are(14)wat edges=0,∂w2∂x2at edges=∂w2∂y2at edges=0.More concretely, the corresponding displacement and stress boundary conditions of every edges are(15)w0,y,t=0,∂2w∂x20,y,t=0,wa,y,t=0,∂2w∂x2a,y,t=0,wx,0,t=0,∂2w∂y2x,0,t=0,wx,b,t=0,∂2w∂y2x,b,t=0,(16)∂2φ∂x20,y,t=0,∂2φ∂x2a,y,t=0,∂2φ∂y2x,0,t=0,∂2φ∂y2x,b,t=0.


## 4. Solution of Fundamental Equations

The functions that satisfy the displacement conditions of every edges equation (15) are separated as follows [12, 13]:(17)wx,y,t=∑m,nTmnt⋅Wmnx,y,where Wmnx,y is the given deformation function; Tmnt is the function of time; m and n are the positive integer.According to the basic vibration theory and boundary conditions, the displacement function is given by(18)Wx,y=sinmπxasinnπyb,where m and n, respectively, denote the orders of vibration displacement in x and y directions.We take one term of equation (18) for computation; i.e.,(19)wx,y,t=Tmntsinmπxasinnπyb.LetWmnx,y=Wx,y=W and Tmnt=Tt=T.The substitution of equation (19) into equation (9) yields(20)1E1∂4φ∂y4+1E2∂4φ∂x4=T2tm2n2π4a2b2cos2mπxa+cos2nπyb+TtWx,yk0xn2π2b2+k0ym2π2a2.The stress functionφx,y,t should satisfy the stress boundary condition (16), on the basic of the differential equation theory. Therefore, the general solution of φx,y,t may be assumed as follows:(21)φx,y,t=T2tαcos2mπxa+βcos2nπyb+γ1x3+γ2x2+γ3x+γ4y3+γ5y2+γ6y+γ7+Ttsinmπxasinnπybλ.Let(22)φx,y,t=T2tΦ1x,y+TtΦ2x,y,Φ1x,y=αcos2mπxa+βcos2nπyb+γ1x3+γ2x2+γ3x+γ4y3+γ5y2+γ6y+γ7,Φ2x,y=λsinmπxasinnπyb=λWx,y.The substitution of equation (21) into equation (16) yields(23)α=E2n2a232m2b2,β=E1m2b232n2a2,λ=k0xnπ/b2+k0ymπ/a2nπ/b4/E1+mπ/a4/E2,γ1=0,γ2=π2E2n216b2,γ4=0,γ5=π2E1m216a2,where γ3,γ6,andγ7 are the arbitrary constant and can be set to zero to simplify the computation according to differential equations theory.Then, the solution ofφx,y,t is(24)φx,y,t=T2tΦ1x,y+TtΦ2x,y,Φ1x,y=E2n2a232m2b2cos2mπxa+E1m2b232n2a2cos2nπyb+π2E2n216b2x2+π2E1m216a2y2,Φ2x,y=k0xnπ/b2+k0ymπ/a2nπ/b4/E1+mπ/a4/E2sinmπxasinnπyb.The substitution of equations (24), (19), and (10) into equation (8) yields(25)ρ0WT″t+cWT′t−k0xh∂2Φ2∂y2+k0yh∂2Φ2∂x2+N0x∂2w∂x2+N0y∂2w∂y2Tt−k0xh∂2Φ1∂y2+k0yh∂2Φ1∂x2+h∂2Φ2∂y2∂2w∂x2+h∂2Φ2∂x2∂2w∂y2T2t−h∂2Φ1∂y2∂2w∂x2+h∂2Φ1∂x2∂2w∂y2T3t=Ftδx−x0y−y0,where T′t=dTt/dtandT″t=d2Tt/dt2.By applying the Bubnov–Galerkin method [4], equation (25) can be transformed into(26)∬sρ0WT″t+cWT′t−k0xh∂2Φ2∂y2+k0yh∂2Φ2∂x2+N0x∂2w∂x2+N0y∂2w∂y2Tt−k0xh∂2Φ1∂y2+k0yh∂2Φ1∂x2+h∂2Φ2∂y2∂2w∂x2+h∂2Φ2∂x2∂2w∂y2T2t−h∂2Φ1∂y2∂2w∂x2+h∂2Φ1∂x2∂2w∂y2T3tWx,ydxdy=∬sFtδx−x0y−y0Wx,ydxdy,where(27)Ft=−M∂2wx0,y0,t∂t2=−MWx0,y0T″t.Equation (26) can be simplified into a homogeneous differential equation as follows:(28)T″t+BA−FT′t+CA−FTt+DA−FT2t+EA−FT3t=0,where(29)A=∬Sρ0W2dxdy=ab4ρ0,B=∬ScW2dxdy=ab4c,C=∬S−k0xh∂2Φ2∂y2+k0yh∂2Φ2∂x2+N0x∂2w∂x2+N0y∂2w∂y2Wdxdy=m2π2b2N0x+hk0yλ+n2π2a2N0y+hk0xλ4ab,D=∬S−k0xh∂2Φ1∂y2+k0yh∂2Φ1∂x2+h∂2Φ2∂y2∂2w∂x2+h∂2Φ2∂x2∂2w∂y2Wdxdy,E=∬S−h∂2Φ1∂y2∂2w∂x2+h∂2Φ1∂x2∂2w∂y2Wdxdy=hπ43E1m4b4+3E2n4a464a3b3,F=∬SFtδx−x0y−y0Wdxdy=−MW2x0,y0=−Msin2mπx0asin2nπy0b.The KBM perturbation method [13] is applied to solve equation (28). Assume the perturbation parameter is ε=h2/ab≪1; then, equation (28) can be simplified by letting x=xt=Tt as follows:(30)x¨+ω02x=εα1x3+α2x2+α3x˙,where(31)ω02=CA−F,α1=−EεA−F,α2=−DεA−F,α3=−BεA−F,x¨=d2xdt2,x˙=dxdt.According to the perturbation method of KBM, letfx,x˙=α1x3+α2x2+α3x˙ and the solution of equation (30) is(32)x=acosψ.In equation (32), a and ψ are determined by(33)dadt=−εω0A0a,dψdt=ω0−εaω0C0a,where(34)A0a=12π∫02πsinφfx,x˙dψ=−12α3aω0,C0a=12π∫02πcosφfx,x˙dψ=38α1a3.The substitution of equaiton (34) into equation (33) yields(35)a=Deεα3t/2,ψ=ω0−3εα1a28ω0t+φ0,where D denotes the amplitude of vibration and φ0 denotes the initial phase of vibration. They can be determined by the initial conditions of membrane vibration.The substitution of equation (35) into equation (32) yields(36)x=xt=Deεα3t/2cosω0−3εα1D2eεα3t8ω0t+φ0.Let(37)ω=ω0−3εα1D2eεα3t8ω0.Equation (36) is the frequency function of vibration. According to equation (36), we can conclude that the amplitude of vibration and damping coefficient of structure have effect on the frequency of the nonlinear damped forced vibration of orthotropic saddle membrane structure.The expression of initial conditions of membrane vibration can be obtained according to the principle of conservation of momentum:(38)xtt=0=0,(39)dxtdtt=0=v0sinmπx0/asinnπy0/b,where v0=MvH/M+4ρHab/π2; vH is the initial velocity of hailstone; v0 is the initial velocity of the system that consisted of hailstone and the membrane impact point; M is the mass of hailstone; ρH is the density of hailstone.The substitution of equation (38) into equation (36) yields(40)x0=Dcosφ0=0.In equation (40), the amplitude of vibration is D>0 and there will undoubtedly be cosφ0=0; therefore, φ0=kπ/2,k=1,3,5,…. Taking φ0=π/2 and substituting it into equation (36) yields(41)x=xt=Deεα3t/2cosω0−3εα1D2eεα3t8ω0t+π2.The substitution of the first derivative of equation (41) into equation (39) yields(42)D3εα1D28ω0−ω0=v0sinmπx0/asinnπy0/b.We solved equation (42) by applying the root formula of simple cubic equation and obtained the real root expression of amplitude:(43)D=481v02α14ε4ω02csc2mπx0/acsc2nπy0/b−32α13ε3ω06+36v0ω0α12ε2cscmπx0/acscnπy0/b33α1ε−481v02α14ε4ω02csc2mπx0/acsc2nπy0/b−32ω06α13ε3−36v0ω0α12ε2cscmπx0/acscnπy0/b33α1ε,where(44)ω0=m2π2b2N0x+hλk0ya2b2ρ0+4abMsin2mπx0/asin2nπy0/b+n2π2a2N0y+hλk0xa2b2ρ0+4abMsin2mπx0/asin2nπy0/b,λ=k0xnπ/b2+k0ymπ/a2nπ/b4/E1+mπ/a4/E2,α1=−3π4E1m4b4+E2n4a416a3b3hρ0+64a2b2hMsin2mπx0/asin2nπy0/b,α3=−a2b2cabρ0h2+4h2Msin2mπx0/asin2nπy0/b,k0x=8f2a2,k0y=−8f1b2,v0=MvHM+4ρHab/π2,ε=h2ab.Whenm+n=4i,i=1,2,3,… and we can only obtain the amplitude of vibration D>0.By substituting of equation (41) into equation (17), the displacement expression is obtained:(45)wx,y,t=∑m=1∞∑n=1∞sinmπxasinnπybDeεα3t/2cosω0−3εα1D2eεα3t8ω0t+π2,where D is determined by equation (43). According to equation (45), we can obtain the mode of displacement of membrane surface.Superposition of the initial surface function of saddle membrane structure equation (1) and its displacement expression equation (45), we can obtain the mode shape of the saddle membrane structure excited by impact of hailstone.(46)S=f2x−a/22a/22−f1y−b/22b/22+∑m=1∞∑n=1∞sinmπxasinnπybDeεα3t/2cosω0−3εα1D2eεα3t8ω0t+π2,where S denotes the mode shape of the saddle membrane structure.According to equation (46), we can obtain the modes of vibration and displacement time histories of the saddle membrane structures.


## 5. Analytical Examples

We take the orthotropic membrane that is widely used in practical engineering application as the analytical example:E1 = 1400 MPa, E2 = 900 MPa [16], the areal density of membrane ρ0 = 1.7 kg/m2, the thickness h = 1.0 mm, the length a = 1.0 m, the width b = 1.0 m, and the viscous damping c = 120 Ns/m. We take the center point of membrane (i.e., x0=a/2, y0=b/2) as the impact point that is excited by hailstone.


### 5.1. Displacement Time Histories of Single Order

The arch-to-span ratiof1 = f2 = 1/10, hailstone diameter d = 5.0 cm, and pretension N = 1000 N/m. According to equation (2), we can obtain k0x=0.8m−1 and k0y=−0.8m−1. According to equation (45), the impact point’s first three-order vibration time histories are shown in Figure 3. From Figure 3, we can conclude that(i)
The maximum vibration amplitude decreases gradually with the increase of the vibration order (i.e., concluded from first order, second order to third order). The maximum vibration amplitude decreases the rate of first order to second order to 16.8% and second order to third order to 8.5%, and the decreases are nonlinear.(ii)
With the increasing of time, the maximum vibration amplitude of single order decreases gradually until it reaches zero.Figure 3
Time history curves of the first three orders of saddle membrane structure: (a) 1st order; (b) 2nd order; (c) 3rd order.
(a)
(b)
(c)


### 5.2. Computation of Amplitude of Impact Point

The arch-to-span ratio are 1/12 and 1/10, the pretension levels increase from 1000 to 4000 N/m, and the hailstone diameters increase from 1.0 to 6.0 cm. According to equation (45), the max. amplitude of the impact point when t = 0 is obtained. The results of the first three orders are presented in Table 2.Table 2
First three-order max. amplitude values (mm) of the impact point.
Arch-to-span ratio
Pretension (N/m)
Order
Hailstone diameter (cm)
1.0
2.0
3.0
4.0
5.0
6.0
1/10
1000
1st
0.016
0.170
0.674
1.783
3.743
7.098
2nd
0.014
0.146
0.575
1.473
3.114
6.042
3rd
0.013
0.140
0.556
1.468
2.849
5.144
2000
1st
0.015
0.160
0.637
1.685
3.543
6.760
2nd
0.013
0.135
0.534
1.415
2.994
5.818
3rd
0.011
0.121
0.479
1.246
2.578
4.767
3000
1st
0.014
0.152
0.604
1.600
3.370
6.461
2nd
0.012
0.130
0.515
1.364
2.886
5.616
3rd
0.010
0.108
0.429
1.132
2.368
4.446
4000
1st
0.014
0.145
0.577
1.527
3.220
6.197
2nd
0.012
0.125
0.497
1.318
2.789
5.433
3rd
0.009
0.100
0.397
1.048
2.201
4.175
1/12
1000
1st
0.019
0.199
0.788
2.079
4.327
8.018
2nd
0.016
0.165
0.655
1.734
3.656
7.026
3rd
0.015
0.160
0.628
1.581
3.169
5.535
2000
1st
0.017
0.184
0.729
1.927
4.030
7.562
2nd
0.015
0.156
0.620
1.642
3.467
6.687
3rd
0.012
0.131
0.519
1.366
2.817
5.102
3000
1st
0.016
0.172
0.682
1.804
3.784
7.165
2nd
0.014
0.149
0.590
1.564
3.303
6.391
3rd
0.011
0.117
0.465
1.227
2.553
4.730
4000
1st
0.015
0.162
0.643
1.701
3.577
6.819
2nd
0.013
0.142
0.565
1.495
3.161
6.129
3rd
0.010
0.107
0.425
1.123
2.349
4.416Figures4–6 show the results of Table 2. According to Table 2 and Figures 4–6, we can come to the following conclusions:(i)
When the arch-to-span ratio is 1/10 and the pretension level is 3000 N/m, the single-order maximum amplitude of the impact point decreases with respect to increasing vibration order; the single-order maximum amplitude of the impact point increases with respect to hailstone diameter increasing, and the increase is nonlinear (i.e., the increment of the maximum amplitude became smaller and smaller).(ii)
When the arch-to-span ratio is 1/10 and the hailstone diameter is 5.0 cm, the single-order maximum amplitude of impact point decreases with respect to the increasing pretensions and the decrement of the maximum amplitude became smaller and smaller. This reflects the nonlinearity of the of the membrane vibration.(iii)
When the pretension level is 3000 N/m and hailstone diameter is 5.0 cm, the impact point maximum amplitude of single order decreases with respect to increasing arch-to-span ratio and the decrease is nonlinear.Figure 4
First three-order max. amplitude under impact of hailstone with different diameters when arch-to-span ratio is 1/10 and pretension level is 3000 N/m.Figure 5
First three-order max. amplitude under different pretension levels when arch-to-span ratio is 1/10 and hailstone diameter is 5.0 cm.Figure 6
First three-order max. amplitude under different arch-to-span ratios when the pretension level is 3000 N/m and hailstone diameter is 5.0 cm.


### 5.3. Total Displacement Time Histories

The arch-to-span ratio is 1/10, and the pretension level is 1000 N/m. According to equation (45), the total displacement time history of impact point is shown in Figure 7, when hailstone diameter is 5.0 cm.Figure 7
Total displacement time history curve of impact point.From Figure7, we can observe that the impact point did weaken the vibration; i.e., the amplitude rapidly increases to its maximum when the membrane is excited by hailstone. Soon afterwards, with the increase of time, the maximum amplitude decreases gradually until it reaches zero.


### 5.4. Computation of Total Displacement of Impact Point

The arch-to-span ratios are 1/12 and 1/10, the pretension levels increase from 1000 to 4000 N/m, and the hailstone diameters increase from 1.0 to 6.0 cm. According to equation (45), the total displacement values of the impact point are calculated and listed in Table 3.Table 3
Max. total displacement values (mm) of the impact point.
Arch-to-span ratio
Pretension (N/m)
Hailstone diameter (cm)
1.0
2.0
3.0
4.0
5.0
6.0
1/10
1000
0.31
2.22
7.36
19.04
37.46
63.06
2000
0.24
1.82
6.16
16.15
32.88
58.25
3000
0.21
1.57
5.05
14.25
29.47
53.87
4000
0.18
1.41
4.73
12.89
25.05
50.11
1/12
1000
0.33
2.39
8.00
20.53
39.64
65.10
2000
0.25
1.90
6.52
17.05
34.45
60.14
3000
0.21
1.63
5.48
14.86
30.63
55.49
4000
0.18
1.45
5.05
13.34
27.77
51.47Figures8–10 show the results of Table 3. According to Table 3 and Figures 8–10, we can come to the following conclusions:(i)
The max. total displacement of the impact point increases with the increase of hailstone diameter. When the arch-to-span ratio is 1/10 and the pretension level is 3000 N/m and with the hailstone diameter increasing from 1.0 to 6.0 cm, the growth rates of maximum total displacement are 648%, 222%, 182%, 107%, and 83%, respectively. The growth rate of the max. total displacement became smaller and smaller.(ii)
As the membrane surface is subjected to effect of stress stiffening, the max. total displacement of impact point decreases with respect to increasing pretension levels. When arch-to-span ratio is 1/10 and the hailstone diameter is 6.0 cm, and with the pretension levels increasing from 1000 to 4000 N/m, the decrease rates of maximum total displacement are 8.3%, 8.1%, and 7.5%, respectively. The decrease rate of the max. total displacement became smaller and smaller. This reflects the nonlinearity of the membrane vibration.(iii)
The max. total displacement of impact point increases with the increase of arch-to-span ratio, and the increase is nonlinear.Figure 8
Max. total displacement curves of the impact point when arch-to-span ratio is 1/10 and the pretension levels increase from 1000 to 4000 N/m.Figure 9
Total displacement curves of the impact point when arch-to-span ratio is 1/12 and the pretension levels increase from 1000 to 4000 N/m.Figure 10
Total displacement curves of the impact point when the pretension level is 3000 N/m and arch-to-span ratio is 1/12 and 1/10.


### 5.5. Computation of Frequency

#### 5.5.1. Hailstone Diameter

The arch-to-span ratio is 1/10, and the pretension level is 3000 N/m. The hailstone diameters increased from 1.0 to 6.0 cm. According to equation (37), the single-order frequency is affected by the hailstone diameter and time. The frequencies of the first three orders with different hailstone diameters and time instants are calculated and listed in Table 4.Table 4
Frequencies (rad/s) under different hailstone diameters and time instants.
Order
Time (s)
Hailstone diameter (cm)
1.0
2.0
3.0
4.0
5.0
6.0
1st order
0.000
767.153
764.210
756.556
743.197
725.732
711.945
0.005
767.153
764.206
756.494
742.780
724.029
706.369
0.010
767.153
764.203
756.449
742.480
722.781
702.175
0.050
767.153
764.197
756.347
741.767
719.633
690.749
0.100
767.153
764.196
756.341
741.714
719.358
689.520
t
⟶
∞
767.153
764.196
756.340
741.712
719.345
689.445
2nd order
0.000
900.943
897.476
888.331
871.663
847.390
819.091
0.005
900.943
897.475
888.306
871.495
846.699
816.760
0.010
900.943
897.474
888.288
871.375
846.192
815.006
0.050
900.943
897.471
888.248
871.088
844.914
810.228
0.100
900.943
897.471
888.245
871.066
844.803
809.714
t
⟶
∞
900.943
897.471
888.245
871.066
844.797
809.683
3rd order
0.000
1081.37
1077.25
1066.77
1049.89
1032.55
1034.530
0.005
1081.37
1077.24
1066.59
1048.66
1027.60
1018.990
0.010
1081.37
1077.23
1066.45
1047.78
1023.97
1007.310
0.050
1081.37
1077.21
1066.15
1045.68
1014.82
975.473
0.100
1081.37
1077.21
1066.13
1045.52
1014.02
972.048
t
⟶
∞
1081.37
1077.21
1066.13
1045.51
1013.98
971.837
4th order
0.000
1105.53
1101.40
1092.06
1083.13
1094.00
1100.19
5th order
0.000
1184.02
1179.48
1167.68
1147.15
1120.50
1165.05
6th order
0.000
1190.68
1186.16
1174.99
1158.63
1147.61
1171.37
7th order
0.000
1395.95
1390.65
1377.50
1358.04
1344.08
1369.30Figures11 and 12 show the results of Table 4. According to Table 4 and Figures 11 and 12, we can come to the following conclusions:(i)
In the case of the hail diameter being constant, the vibration frequencies increase with respect to the increasing vibration order. The frequencies of each order are maximum att = 0 and gradually decrease to ω0 with increase of time. This reflects the nonlinearity of the damped forced vibration of the membrane.(ii)
Whent = 0, according to equation (37), the vibration frequency value is dependent on ω0 and amplitude. Amplitudes of each order increase with the increase of hailstone diameter. This shows that the vibration frequency value of t = 0 decreases with respect to increasing hailstone diameter. But, the higher-order vibration (i.e., begin with the 3rd order) frequencies value of t = 0 increase when hailstone diameter increases from 5.0 to 6.0 cm. According to equation (43) and equation (38), in terms of the velocity and the mass of hailstone, the velocity has a higher-order (i.e., v2) effect on vibration amplitude and frequency than the mass (i.e. m−1), which is more significant in higher-order vibration. When hailstone diameter increased from 5.0 to 6.0 cm, as shown in Table 1, the velocity squared difference increases rather than decreasing. Namely, the increase of velocity determines the increase of frequency as hailstone diameter increased from 5.0 to 6.0 cm. This is the reason why the higher-order vibration frequency value of t = 0 increases when hailstone diameter increased from 5.0 to 6.0 cm.(iii)
According to equation (37), when t⟶∞, the vibration frequency value is only dependent on ω0. The vibration frequency value of each order decreases with respect to increasing hailstone diameter.Figure 11
Frequencies (rad/s) under different hailstone diameters and time instants.Figure 12
Frequencies (rad/s) under different hailstone diameters att = 0.


#### 5.5.2. Pretension Levels

The arch-to-span ratio is 1/10, and the hailstone diameter is 5.0 cm. The pretension levels increased from 1000 to 4000 N/m. Table5 shows the first three-order frequencies with different pretension levels and times, which are calculated according to equation (37).Table 5
Frequencies (rad/s) under different pretensions and time instants.
Order
Time (s)
Pretension (N/m)
1000
2000
3000
4000
1st order
0.000
653.370
690.417
725.732
759.517
0.005
651.026
688.436
724.029
758.034
0.010
649.307
686.983
722.781
756.947
0.050
644.972
683.319
719.633
754.205
0.100
644.594
682.999
719.358
753.965
t
⟶
∞
644.576
682.984
719.345
753.954
2nd order
0.000
785.375
816.955
847.390
876.795
0.005
784.506
816.183
846.699
876.171
0.010
783.869
815.617
846.192
875.714
0.050
782.262
814.190
844.914
874.561
0.100
782.122
814.066
844.803
874.460
t
⟶
∞
782.115
814.060
844.797
874.455
3rd order
0.000
858.477
948.671
1032.55
1110.97
0.005
849.685
942.243
1027.60
1107.02
0.010
843.237
937.528
1023.97
1104.12
0.050
826.978
925.641
1014.82
1096.81
0.100
825.559
924.603
1014.02
1096.17
t
⟶
∞
825.492
924.554
1013.98
1096.14Figure13 shows the results of Table 5. According to Table 5 and Figure 13, we can conclude that the single-order frequencies increase with the increasing pretension levels. Moreover, the frequency increment of each order is bigger and bigger, so the increase is nonlinear.Figure 13
Frequencies (rad/s) under different pretension levels and time instants.


#### 5.5.3. Arch-to-Span Ratio

The pretension level is 3000 N/m, and the hailstone diameter is 5.0 cm. The arch-to-span ratio increased from 1/12 to 1/10. According to equation (37), the first three-order frequencies with different arch-to-span ratios and times are calculated and presented in Table 6.Table 6
Frequencies (rad/s) under different arch-to-span ratios and time instants.
Order
Time (s)
Arch-to-span ratio
1/10
1/12
1st order
0.000
725.732
646.338
0.005
724.029
643.915
0.010
722.781
642.138
0.050
719.633
637.658
0.100
719.358
637.267
t
⟶
∞
719.345
637.248
2nd order
0.000
847.390
740.343
0.005
846.699
739.304
0.010
846.192
738.543
0.050
844.914
736.622
0.100
844.803
736.454
t
⟶
∞
844.797
736.446
3rd order
0.000
1032.55
957.718
0.005
1027.60
951.477
0.010
1023.97
946.899
0.050
1014.82
935.357
0.100
1014.02
934.349
t
⟶
∞
1013.98
934.302Figure14 shows the results of Table 6. According to Table 6 and Figure 14, we can conclude that the vibration frequencies of each order decrease with the decreasing arch-to-span ratio.Figure 14
Frequencies (rad/s) with different arch-to-span ratios and time instants.


### 5.6. Mode Shape

The hailstone diameter is 6.0 cm, arch-to-span ratio is 1/10, pretension level is 1000 N/m, and timet = 0.002 s and 0.007 s. According to equation (46), the first three-order mode shapes are presented in Figures 15–20. By superimposing the first three-order mode shapes, the superposed mode shapes are obtained and shown in Figures 21 and 22. The coordinate dimension of Figures 15–22 is in meter.Figure 15
First order (t = 0.002 s).Figure 16
First order (t = 0.007 s).Figure 17
Second order (t = 0.002 s).Figure 18
Second order (t = 0.007 s).Figure 19
Third order (t = 0.002 s).Figure 20
Third order (t = 0.007 s).Figure 21
Superposing of the first three orders (t = 0.002 s).Figure 22
Superposing of the first three orders (t = 0.007 s).By analyzing the mode shapes, we can conclude that the mode shape function equation (46) can be applied to calculate the single-order mode shapes and the total superposed mode shapes of the damped large nonlinear deflection vibration of orthotropic saddle membrane structures excited by hailstone impact load succinctly. In practical engineering vibration, the low-order mode shapes have a much bigger affect than the higher-order mode shapes, so we take the first three-order mode shapes into analysis.


### 5.7. Brief Summary

(1)
The arch-to-span ratio is 1/10, and the pretension level is 3000 N/m. The hailstone diameters increased from 1.0 to 6.0 cm. We plot the first trough of time histories of the first-order vibration in Figure23(a). From Figure 23(a), we can conclude that the hailstone diameter increase has a significant effect on the increase of amplitude and small effect on the decrease of frequency and increase of period.
(2)
The hailstone diameter is 5.0 cm, and the arch-to-span ratios are 1/12 and 1/10. The pretension levels increased from 1000 to 4000 N/m. We plot the first trough of time histories of the first-order vibration in Figure23(b). From Figure 23(b), we can conclude that the arch-to-span ratio decrease has a more significant effect on the increase of amplitude and the increase of period than that of decreasing pretension.Figure 23
First trough of time histories of first-order vibration: (a) the arch-to-span ratio is 1/10, pretension level is 3000 N/m, and hailstone diameters increased from 1.0 to 6.0 cm (D1 to D6); (b) the hailstone diameter is 5.0 cm, the arch-to-span ratio are 1/12 and 1/10 (R10 and R12), and pretension levels increased from 1000 to 4000 N/m (N1000 to N4000).
(a)
(b)


## 5.1. Displacement Time Histories of Single Order

The arch-to-span ratiof1 = f2 = 1/10, hailstone diameter d = 5.0 cm, and pretension N = 1000 N/m. According to equation (2), we can obtain k0x=0.8m−1 and k0y=−0.8m−1. According to equation (45), the impact point’s first three-order vibration time histories are shown in Figure 3. From Figure 3, we can conclude that(i)
The maximum vibration amplitude decreases gradually with the increase of the vibration order (i.e., concluded from first order, second order to third order). The maximum vibration amplitude decreases the rate of first order to second order to 16.8% and second order to third order to 8.5%, and the decreases are nonlinear.(ii)
With the increasing of time, the maximum vibration amplitude of single order decreases gradually until it reaches zero.Figure 3
Time history curves of the first three orders of saddle membrane structure: (a) 1st order; (b) 2nd order; (c) 3rd order.
(a)
(b)
(c)


## 5.2. Computation of Amplitude of Impact Point

The arch-to-span ratio are 1/12 and 1/10, the pretension levels increase from 1000 to 4000 N/m, and the hailstone diameters increase from 1.0 to 6.0 cm. According to equation (45), the max. amplitude of the impact point when t = 0 is obtained. The results of the first three orders are presented in Table 2.Table 2
First three-order max. amplitude values (mm) of the impact point.
Arch-to-span ratio
Pretension (N/m)
Order
Hailstone diameter (cm)
1.0
2.0
3.0
4.0
5.0
6.0
1/10
1000
1st
0.016
0.170
0.674
1.783
3.743
7.098
2nd
0.014
0.146
0.575
1.473
3.114
6.042
3rd
0.013
0.140
0.556
1.468
2.849
5.144
2000
1st
0.015
0.160
0.637
1.685
3.543
6.760
2nd
0.013
0.135
0.534
1.415
2.994
5.818
3rd
0.011
0.121
0.479
1.246
2.578
4.767
3000
1st
0.014
0.152
0.604
1.600
3.370
6.461
2nd
0.012
0.130
0.515
1.364
2.886
5.616
3rd
0.010
0.108
0.429
1.132
2.368
4.446
4000
1st
0.014
0.145
0.577
1.527
3.220
6.197
2nd
0.012
0.125
0.497
1.318
2.789
5.433
3rd
0.009
0.100
0.397
1.048
2.201
4.175
1/12
1000
1st
0.019
0.199
0.788
2.079
4.327
8.018
2nd
0.016
0.165
0.655
1.734
3.656
7.026
3rd
0.015
0.160
0.628
1.581
3.169
5.535
2000
1st
0.017
0.184
0.729
1.927
4.030
7.562
2nd
0.015
0.156
0.620
1.642
3.467
6.687
3rd
0.012
0.131
0.519
1.366
2.817
5.102
3000
1st
0.016
0.172
0.682
1.804
3.784
7.165
2nd
0.014
0.149
0.590
1.564
3.303
6.391
3rd
0.011
0.117
0.465
1.227
2.553
4.730
4000
1st
0.015
0.162
0.643
1.701
3.577
6.819
2nd
0.013
0.142
0.565
1.495
3.161
6.129
3rd
0.010
0.107
0.425
1.123
2.349
4.416Figures4–6 show the results of Table 2. According to Table 2 and Figures 4–6, we can come to the following conclusions:(i)
When the arch-to-span ratio is 1/10 and the pretension level is 3000 N/m, the single-order maximum amplitude of the impact point decreases with respect to increasing vibration order; the single-order maximum amplitude of the impact point increases with respect to hailstone diameter increasing, and the increase is nonlinear (i.e., the increment of the maximum amplitude became smaller and smaller).(ii)
When the arch-to-span ratio is 1/10 and the hailstone diameter is 5.0 cm, the single-order maximum amplitude of impact point decreases with respect to the increasing pretensions and the decrement of the maximum amplitude became smaller and smaller. This reflects the nonlinearity of the of the membrane vibration.(iii)
When the pretension level is 3000 N/m and hailstone diameter is 5.0 cm, the impact point maximum amplitude of single order decreases with respect to increasing arch-to-span ratio and the decrease is nonlinear.Figure 4
First three-order max. amplitude under impact of hailstone with different diameters when arch-to-span ratio is 1/10 and pretension level is 3000 N/m.Figure 5
First three-order max. amplitude under different pretension levels when arch-to-span ratio is 1/10 and hailstone diameter is 5.0 cm.Figure 6
First three-order max. amplitude under different arch-to-span ratios when the pretension level is 3000 N/m and hailstone diameter is 5.0 cm.


## 5.3. Total Displacement Time Histories

The arch-to-span ratio is 1/10, and the pretension level is 1000 N/m. According to equation (45), the total displacement time history of impact point is shown in Figure 7, when hailstone diameter is 5.0 cm.Figure 7
Total displacement time history curve of impact point.From Figure7, we can observe that the impact point did weaken the vibration; i.e., the amplitude rapidly increases to its maximum when the membrane is excited by hailstone. Soon afterwards, with the increase of time, the maximum amplitude decreases gradually until it reaches zero.


## 5.4. Computation of Total Displacement of Impact Point

The arch-to-span ratios are 1/12 and 1/10, the pretension levels increase from 1000 to 4000 N/m, and the hailstone diameters increase from 1.0 to 6.0 cm. According to equation (45), the total displacement values of the impact point are calculated and listed in Table 3.Table 3
Max. total displacement values (mm) of the impact point.
Arch-to-span ratio
Pretension (N/m)
Hailstone diameter (cm)
1.0
2.0
3.0
4.0
5.0
6.0
1/10
1000
0.31
2.22
7.36
19.04
37.46
63.06
2000
0.24
1.82
6.16
16.15
32.88
58.25
3000
0.21
1.57
5.05
14.25
29.47
53.87
4000
0.18
1.41
4.73
12.89
25.05
50.11
1/12
1000
0.33
2.39
8.00
20.53
39.64
65.10
2000
0.25
1.90
6.52
17.05
34.45
60.14
3000
0.21
1.63
5.48
14.86
30.63
55.49
4000
0.18
1.45
5.05
13.34
27.77
51.47Figures8–10 show the results of Table 3. According to Table 3 and Figures 8–10, we can come to the following conclusions:(i)
The max. total displacement of the impact point increases with the increase of hailstone diameter. When the arch-to-span ratio is 1/10 and the pretension level is 3000 N/m and with the hailstone diameter increasing from 1.0 to 6.0 cm, the growth rates of maximum total displacement are 648%, 222%, 182%, 107%, and 83%, respectively. The growth rate of the max. total displacement became smaller and smaller.(ii)
As the membrane surface is subjected to effect of stress stiffening, the max. total displacement of impact point decreases with respect to increasing pretension levels. When arch-to-span ratio is 1/10 and the hailstone diameter is 6.0 cm, and with the pretension levels increasing from 1000 to 4000 N/m, the decrease rates of maximum total displacement are 8.3%, 8.1%, and 7.5%, respectively. The decrease rate of the max. total displacement became smaller and smaller. This reflects the nonlinearity of the membrane vibration.(iii)
The max. total displacement of impact point increases with the increase of arch-to-span ratio, and the increase is nonlinear.Figure 8
Max. total displacement curves of the impact point when arch-to-span ratio is 1/10 and the pretension levels increase from 1000 to 4000 N/m.Figure 9
Total displacement curves of the impact point when arch-to-span ratio is 1/12 and the pretension levels increase from 1000 to 4000 N/m.Figure 10
Total displacement curves of the impact point when the pretension level is 3000 N/m and arch-to-span ratio is 1/12 and 1/10.


## 5.5. Computation of Frequency

### 5.5.1. Hailstone Diameter

The arch-to-span ratio is 1/10, and the pretension level is 3000 N/m. The hailstone diameters increased from 1.0 to 6.0 cm. According to equation (37), the single-order frequency is affected by the hailstone diameter and time. The frequencies of the first three orders with different hailstone diameters and time instants are calculated and listed in Table 4.Table 4
Frequencies (rad/s) under different hailstone diameters and time instants.
Order
Time (s)
Hailstone diameter (cm)
1.0
2.0
3.0
4.0
5.0
6.0
1st order
0.000
767.153
764.210
756.556
743.197
725.732
711.945
0.005
767.153
764.206
756.494
742.780
724.029
706.369
0.010
767.153
764.203
756.449
742.480
722.781
702.175
0.050
767.153
764.197
756.347
741.767
719.633
690.749
0.100
767.153
764.196
756.341
741.714
719.358
689.520
t
⟶
∞
767.153
764.196
756.340
741.712
719.345
689.445
2nd order
0.000
900.943
897.476
888.331
871.663
847.390
819.091
0.005
900.943
897.475
888.306
871.495
846.699
816.760
0.010
900.943
897.474
888.288
871.375
846.192
815.006
0.050
900.943
897.471
888.248
871.088
844.914
810.228
0.100
900.943
897.471
888.245
871.066
844.803
809.714
t
⟶
∞
900.943
897.471
888.245
871.066
844.797
809.683
3rd order
0.000
1081.37
1077.25
1066.77
1049.89
1032.55
1034.530
0.005
1081.37
1077.24
1066.59
1048.66
1027.60
1018.990
0.010
1081.37
1077.23
1066.45
1047.78
1023.97
1007.310
0.050
1081.37
1077.21
1066.15
1045.68
1014.82
975.473
0.100
1081.37
1077.21
1066.13
1045.52
1014.02
972.048
t
⟶
∞
1081.37
1077.21
1066.13
1045.51
1013.98
971.837
4th order
0.000
1105.53
1101.40
1092.06
1083.13
1094.00
1100.19
5th order
0.000
1184.02
1179.48
1167.68
1147.15
1120.50
1165.05
6th order
0.000
1190.68
1186.16
1174.99
1158.63
1147.61
1171.37
7th order
0.000
1395.95
1390.65
1377.50
1358.04
1344.08
1369.30Figures11 and 12 show the results of Table 4. According to Table 4 and Figures 11 and 12, we can come to the following conclusions:(i)
In the case of the hail diameter being constant, the vibration frequencies increase with respect to the increasing vibration order. The frequencies of each order are maximum att = 0 and gradually decrease to ω0 with increase of time. This reflects the nonlinearity of the damped forced vibration of the membrane.(ii)
Whent = 0, according to equation (37), the vibration frequency value is dependent on ω0 and amplitude. Amplitudes of each order increase with the increase of hailstone diameter. This shows that the vibration frequency value of t = 0 decreases with respect to increasing hailstone diameter. But, the higher-order vibration (i.e., begin with the 3rd order) frequencies value of t = 0 increase when hailstone diameter increases from 5.0 to 6.0 cm. According to equation (43) and equation (38), in terms of the velocity and the mass of hailstone, the velocity has a higher-order (i.e., v2) effect on vibration amplitude and frequency than the mass (i.e. m−1), which is more significant in higher-order vibration. When hailstone diameter increased from 5.0 to 6.0 cm, as shown in Table 1, the velocity squared difference increases rather than decreasing. Namely, the increase of velocity determines the increase of frequency as hailstone diameter increased from 5.0 to 6.0 cm. This is the reason why the higher-order vibration frequency value of t = 0 increases when hailstone diameter increased from 5.0 to 6.0 cm.(iii)
According to equation (37), when t⟶∞, the vibration frequency value is only dependent on ω0. The vibration frequency value of each order decreases with respect to increasing hailstone diameter.Figure 11
Frequencies (rad/s) under different hailstone diameters and time instants.Figure 12
Frequencies (rad/s) under different hailstone diameters att = 0.


### 5.5.2. Pretension Levels

The arch-to-span ratio is 1/10, and the hailstone diameter is 5.0 cm. The pretension levels increased from 1000 to 4000 N/m. Table5 shows the first three-order frequencies with different pretension levels and times, which are calculated according to equation (37).Table 5
Frequencies (rad/s) under different pretensions and time instants.
Order
Time (s)
Pretension (N/m)
1000
2000
3000
4000
1st order
0.000
653.370
690.417
725.732
759.517
0.005
651.026
688.436
724.029
758.034
0.010
649.307
686.983
722.781
756.947
0.050
644.972
683.319
719.633
754.205
0.100
644.594
682.999
719.358
753.965
t
⟶
∞
644.576
682.984
719.345
753.954
2nd order
0.000
785.375
816.955
847.390
876.795
0.005
784.506
816.183
846.699
876.171
0.010
783.869
815.617
846.192
875.714
0.050
782.262
814.190
844.914
874.561
0.100
782.122
814.066
844.803
874.460
t
⟶
∞
782.115
814.060
844.797
874.455
3rd order
0.000
858.477
948.671
1032.55
1110.97
0.005
849.685
942.243
1027.60
1107.02
0.010
843.237
937.528
1023.97
1104.12
0.050
826.978
925.641
1014.82
1096.81
0.100
825.559
924.603
1014.02
1096.17
t
⟶
∞
825.492
924.554
1013.98
1096.14Figure13 shows the results of Table 5. According to Table 5 and Figure 13, we can conclude that the single-order frequencies increase with the increasing pretension levels. Moreover, the frequency increment of each order is bigger and bigger, so the increase is nonlinear.Figure 13
Frequencies (rad/s) under different pretension levels and time instants.


### 5.5.3. Arch-to-Span Ratio

The pretension level is 3000 N/m, and the hailstone diameter is 5.0 cm. The arch-to-span ratio increased from 1/12 to 1/10. According to equation (37), the first three-order frequencies with different arch-to-span ratios and times are calculated and presented in Table 6.Table 6
Frequencies (rad/s) under different arch-to-span ratios and time instants.
Order
Time (s)
Arch-to-span ratio
1/10
1/12
1st order
0.000
725.732
646.338
0.005
724.029
643.915
0.010
722.781
642.138
0.050
719.633
637.658
0.100
719.358
637.267
t
⟶
∞
719.345
637.248
2nd order
0.000
847.390
740.343
0.005
846.699
739.304
0.010
846.192
738.543
0.050
844.914
736.622
0.100
844.803
736.454
t
⟶
∞
844.797
736.446
3rd order
0.000
1032.55
957.718
0.005
1027.60
951.477
0.010
1023.97
946.899
0.050
1014.82
935.357
0.100
1014.02
934.349
t
⟶
∞
1013.98
934.302Figure14 shows the results of Table 6. According to Table 6 and Figure 14, we can conclude that the vibration frequencies of each order decrease with the decreasing arch-to-span ratio.Figure 14
Frequencies (rad/s) with different arch-to-span ratios and time instants.


## 5.5.1. Hailstone Diameter

The arch-to-span ratio is 1/10, and the pretension level is 3000 N/m. The hailstone diameters increased from 1.0 to 6.0 cm. According to equation (37), the single-order frequency is affected by the hailstone diameter and time. The frequencies of the first three orders with different hailstone diameters and time instants are calculated and listed in Table 4.Table 4
Frequencies (rad/s) under different hailstone diameters and time instants.
Order
Time (s)
Hailstone diameter (cm)
1.0
2.0
3.0
4.0
5.0
6.0
1st order
0.000
767.153
764.210
756.556
743.197
725.732
711.945
0.005
767.153
764.206
756.494
742.780
724.029
706.369
0.010
767.153
764.203
756.449
742.480
722.781
702.175
0.050
767.153
764.197
756.347
741.767
719.633
690.749
0.100
767.153
764.196
756.341
741.714
719.358
689.520
t
⟶
∞
767.153
764.196
756.340
741.712
719.345
689.445
2nd order
0.000
900.943
897.476
888.331
871.663
847.390
819.091
0.005
900.943
897.475
888.306
871.495
846.699
816.760
0.010
900.943
897.474
888.288
871.375
846.192
815.006
0.050
900.943
897.471
888.248
871.088
844.914
810.228
0.100
900.943
897.471
888.245
871.066
844.803
809.714
t
⟶
∞
900.943
897.471
888.245
871.066
844.797
809.683
3rd order
0.000
1081.37
1077.25
1066.77
1049.89
1032.55
1034.530
0.005
1081.37
1077.24
1066.59
1048.66
1027.60
1018.990
0.010
1081.37
1077.23
1066.45
1047.78
1023.97
1007.310
0.050
1081.37
1077.21
1066.15
1045.68
1014.82
975.473
0.100
1081.37
1077.21
1066.13
1045.52
1014.02
972.048
t
⟶
∞
1081.37
1077.21
1066.13
1045.51
1013.98
971.837
4th order
0.000
1105.53
1101.40
1092.06
1083.13
1094.00
1100.19
5th order
0.000
1184.02
1179.48
1167.68
1147.15
1120.50
1165.05
6th order
0.000
1190.68
1186.16
1174.99
1158.63
1147.61
1171.37
7th order
0.000
1395.95
1390.65
1377.50
1358.04
1344.08
1369.30Figures11 and 12 show the results of Table 4. According to Table 4 and Figures 11 and 12, we can come to the following conclusions:(i)
In the case of the hail diameter being constant, the vibration frequencies increase with respect to the increasing vibration order. The frequencies of each order are maximum att = 0 and gradually decrease to ω0 with increase of time. This reflects the nonlinearity of the damped forced vibration of the membrane.(ii)
Whent = 0, according to equation (37), the vibration frequency value is dependent on ω0 and amplitude. Amplitudes of each order increase with the increase of hailstone diameter. This shows that the vibration frequency value of t = 0 decreases with respect to increasing hailstone diameter. But, the higher-order vibration (i.e., begin with the 3rd order) frequencies value of t = 0 increase when hailstone diameter increases from 5.0 to 6.0 cm. According to equation (43) and equation (38), in terms of the velocity and the mass of hailstone, the velocity has a higher-order (i.e., v2) effect on vibration amplitude and frequency than the mass (i.e. m−1), which is more significant in higher-order vibration. When hailstone diameter increased from 5.0 to 6.0 cm, as shown in Table 1, the velocity squared difference increases rather than decreasing. Namely, the increase of velocity determines the increase of frequency as hailstone diameter increased from 5.0 to 6.0 cm. This is the reason why the higher-order vibration frequency value of t = 0 increases when hailstone diameter increased from 5.0 to 6.0 cm.(iii)
According to equation (37), when t⟶∞, the vibration frequency value is only dependent on ω0. The vibration frequency value of each order decreases with respect to increasing hailstone diameter.Figure 11
Frequencies (rad/s) under different hailstone diameters and time instants.Figure 12
Frequencies (rad/s) under different hailstone diameters att = 0.


## 5.5.2. Pretension Levels

The arch-to-span ratio is 1/10, and the hailstone diameter is 5.0 cm. The pretension levels increased from 1000 to 4000 N/m. Table5 shows the first three-order frequencies with different pretension levels and times, which are calculated according to equation (37).Table 5
Frequencies (rad/s) under different pretensions and time instants.
Order
Time (s)
Pretension (N/m)
1000
2000
3000
4000
1st order
0.000
653.370
690.417
725.732
759.517
0.005
651.026
688.436
724.029
758.034
0.010
649.307
686.983
722.781
756.947
0.050
644.972
683.319
719.633
754.205
0.100
644.594
682.999
719.358
753.965
t
⟶
∞
644.576
682.984
719.345
753.954
2nd order
0.000
785.375
816.955
847.390
876.795
0.005
784.506
816.183
846.699
876.171
0.010
783.869
815.617
846.192
875.714
0.050
782.262
814.190
844.914
874.561
0.100
782.122
814.066
844.803
874.460
t
⟶
∞
782.115
814.060
844.797
874.455
3rd order
0.000
858.477
948.671
1032.55
1110.97
0.005
849.685
942.243
1027.60
1107.02
0.010
843.237
937.528
1023.97
1104.12
0.050
826.978
925.641
1014.82
1096.81
0.100
825.559
924.603
1014.02
1096.17
t
⟶
∞
825.492
924.554
1013.98
1096.14Figure13 shows the results of Table 5. According to Table 5 and Figure 13, we can conclude that the single-order frequencies increase with the increasing pretension levels. Moreover, the frequency increment of each order is bigger and bigger, so the increase is nonlinear.Figure 13
Frequencies (rad/s) under different pretension levels and time instants.


## 5.5.3. Arch-to-Span Ratio

The pretension level is 3000 N/m, and the hailstone diameter is 5.0 cm. The arch-to-span ratio increased from 1/12 to 1/10. According to equation (37), the first three-order frequencies with different arch-to-span ratios and times are calculated and presented in Table 6.Table 6
Frequencies (rad/s) under different arch-to-span ratios and time instants.
Order
Time (s)
Arch-to-span ratio
1/10
1/12
1st order
0.000
725.732
646.338
0.005
724.029
643.915
0.010
722.781
642.138
0.050
719.633
637.658
0.100
719.358
637.267
t
⟶
∞
719.345
637.248
2nd order
0.000
847.390
740.343
0.005
846.699
739.304
0.010
846.192
738.543
0.050
844.914
736.622
0.100
844.803
736.454
t
⟶
∞
844.797
736.446
3rd order
0.000
1032.55
957.718
0.005
1027.60
951.477
0.010
1023.97
946.899
0.050
1014.82
935.357
0.100
1014.02
934.349
t
⟶
∞
1013.98
934.302Figure14 shows the results of Table 6. According to Table 6 and Figure 14, we can conclude that the vibration frequencies of each order decrease with the decreasing arch-to-span ratio.Figure 14
Frequencies (rad/s) with different arch-to-span ratios and time instants.


## 5.6. Mode Shape

The hailstone diameter is 6.0 cm, arch-to-span ratio is 1/10, pretension level is 1000 N/m, and timet = 0.002 s and 0.007 s. According to equation (46), the first three-order mode shapes are presented in Figures 15–20. By superimposing the first three-order mode shapes, the superposed mode shapes are obtained and shown in Figures 21 and 22. The coordinate dimension of Figures 15–22 is in meter.Figure 15
First order (t = 0.002 s).Figure 16
First order (t = 0.007 s).Figure 17
Second order (t = 0.002 s).Figure 18
Second order (t = 0.007 s).Figure 19
Third order (t = 0.002 s).Figure 20
Third order (t = 0.007 s).Figure 21
Superposing of the first three orders (t = 0.002 s).Figure 22
Superposing of the first three orders (t = 0.007 s).By analyzing the mode shapes, we can conclude that the mode shape function equation (46) can be applied to calculate the single-order mode shapes and the total superposed mode shapes of the damped large nonlinear deflection vibration of orthotropic saddle membrane structures excited by hailstone impact load succinctly. In practical engineering vibration, the low-order mode shapes have a much bigger affect than the higher-order mode shapes, so we take the first three-order mode shapes into analysis.


## 5.7. Brief Summary

(1)
The arch-to-span ratio is 1/10, and the pretension level is 3000 N/m. The hailstone diameters increased from 1.0 to 6.0 cm. We plot the first trough of time histories of the first-order vibration in Figure23(a). From Figure 23(a), we can conclude that the hailstone diameter increase has a significant effect on the increase of amplitude and small effect on the decrease of frequency and increase of period.
(2)
The hailstone diameter is 5.0 cm, and the arch-to-span ratios are 1/12 and 1/10. The pretension levels increased from 1000 to 4000 N/m. We plot the first trough of time histories of the first-order vibration in Figure23(b). From Figure 23(b), we can conclude that the arch-to-span ratio decrease has a more significant effect on the increase of amplitude and the increase of period than that of decreasing pretension.Figure 23
First trough of time histories of first-order vibration: (a) the arch-to-span ratio is 1/10, pretension level is 3000 N/m, and hailstone diameters increased from 1.0 to 6.0 cm (D1 to D6); (b) the hailstone diameter is 5.0 cm, the arch-to-span ratio are 1/12 and 1/10 (R10 and R12), and pretension levels increased from 1000 to 4000 N/m (N1000 to N4000).
(a)
(b)


## 6. Numerical Simulation

In this section, we apply the universal explicit dynamics finite element analysis software ANSYS/LS-DYNA to simulate the process of hailstone impacting on the membrane based on the explicit-to-implicit sequential solution method. In implicit computation, Shell181 element is applied for the membrane and Solid185 element is applied for the hailstone. In explicit dynamic analysis, the elements will be converted to Shell163 and Solid164 element accordingly. We adopt a mapping triangle-shaped shell element to generate the mesh of the membrane surfaces and a mapping hexahedral-shaped solid element to generate the mesh for the hailstone. In ANSYS/LS-Dyna, the simulation results can converge better by refining the finite element mesh [33]. When hailstone diameter is 6.0 cm and arch-to-span ratio is 1/10, 5408 shell elements are generated with element size 2 cm and 4000 solid elements are generated with element size 0.5 cm, which satisfy the convergence accuracy. The results of the meshing are shown in Figures 24 and 25. The pretension load 1000 N/m is applied to the shell elements in implicit computation and converted to explicit dynamic analysis. In explicit dynamic analysis, the velocity according to Table 1 is applied to hailstone, automatic surface-to-surface contact is defined, and the corresponding parameter of the contact surface is set to describe the complex interaction among membranes in the large deformation contact and dynamic impact of hailstone. The Rayleigh damping coefficient is set to 0.05 that is recommended in explicit dynamic analysis.Figure 24
Solid elements of hailstone (diameter is 6.0 cm).Figure 25
Shell of elements of membrane (arch-to-span ratio is 1/10).Dynamic response process of hailstone impacting on membrane is shown in Figure26. Time histories of the vertical displacement of impact point are shown in Figure 27. The following conclusions are drawn based on Figures 26 and 27:(i)
At the initial time (t = 0), the stress distribution in the membrane is more uniform. The maximum stress appears near the highest point and the lowest point and the minimum stress occurs near four corners. The mean stress in the membrane approximately is 1000 N/m.(ii)
The membrane material is elastic; it can only dissipate energy by deforming after being subjected to impact load. Therefore, when the hailstone impacted the membrane surface, most of the energy of hailstone is converted into the membrane strain energy and the kinetic energy of the vibration [12], which results in the rapid increment of vertical displacement of the impact point. At the same time, the stress concentration occurs in the impact area and the membrane stress diffuses from the impact point to the two high points.(iii)
When the displacement of impact point is maximized, because of the pretension, the hailstone is rebounded up from the membrane and the membrane begins to vibrate freely. Subsequently, the displacement of the membrane attenuates gradually, and the vibration energy wave diffuses from the center to all around but rebounds at the boundary of membrane. This is the reason why the membrane stress exhibited irregularity.Figure 26
Dynamic response process of hailstone impacting on membrane surface: (a) when time = 0; (b) when hailstone impact on membrane surface; (c) and (d) when the displacement of the membrane is maximum; (e) and (f) when hailstone is bounced up.
(a)
(b)
(c)
(d)
(e)
(f)Figure 27
Time histories of the vertical displacement of impact point.


## 7. Comparison and Analysis of Numerical Simulation and Theoretical Solution

The comparison of numerical simulation results and the theoretical results shows that the simulation results tally with the actual theoretical situation.Figure28 shows the results of Table 7. According to Table 7 and Figure 28, we can come to the following conclusions:(i)
The nonlinear dynamic response law of orthotropic saddle membrane structures excited by hailstone impact load which reflected in the numerical simulation results is consistent with the theoretical ones.(ii)
We neglected the dead load of membrane in the theoretical calculation. As a result of this, the theoretical results are slightly smaller (the maximum difference is less than 2.63 mm) than those of numerical simulation. The dead load of membrane is much more than the impact load of a hailstone when the diameters of hailstone are low; this is the main reason why the relative errors of simulation and theory are high at low hailstone diameters.(iii)
The theoretical calculation results basically fit the numerical simulation results, implying that our methods have been successful.Figure 28
Comparison of impact point maximum displacement results between numerical simulations and theoretical solutions.Table 7
Comparison of impact point maximum displacement results (mm) between numerical simulation and theoretical solution.
Results
Arch-to-span ratio
Pretension (N/m)
Hailstone diameter (cm)
1.0
2.0
3.0
4.0
5.0
6.0
Simulation
1/10
1000
1.77
4.69
9.18
20.48
38.90
64.48
Theory
1/10
1000
0.31
2.22
7.36
19.04
37.46
63.06
Relative error
82%
53%
20%
7%
4%
2%
Simulation
1/12
1000
2.73
5.02
9.42
20.97
40.14
66.53
Theory
1/12
1000
0.33
2.39
8.00
20.53
39.64
65.10
Relative error
88%
52%
15%
2%
1%
2%


## 8. Conclusions

In this paper, the approximate formulas of hailstone terminal velocity were substituted into the governing equations of the large deflection nonlinear damped vibration of orthotropic saddle membrane structures excited by impact load. And, solving the governing equations by applying the Bubnov–Galerkin method and the method of KBM perturbation, the approximate theoretical solution of the frequency function and displacement function of the large deflection nonlinear damped vibration of saddle membrane structures with four edges simply supported excited by hailstone impact was obtained.The analytical examples proved that the mode shape function equation (46) can be applied to calculate the single-order mode shapes and the total superposed mode shapes of the damped large nonlinear deflection vibration of orthotropic saddle membrane structures excited by hailstone impact load succinctly. In addition, we compare and analyze the results of vibration frequency, amplitude, time histories, and total displacement of membrane structures with different pretensions and arch-to-span ratios under the impact of different size hailstones, and the following conclusions can be drawn:(i)
The increasing hailstone diameter has a significant effect on the increase of amplitude and small effect on the decrease of frequency and increase of period(ii)
The decreasing arch-to-span ratio has a more significant effect on the increase of amplitude and the increase of period than that of decreasing pretensionThe correctness of the analytical theory is verified by comparing with the results of numerical simulation. In addition, the results of this paper can be applied in computation for the vibration control and dynamic design of practical spatial membrane structure under impact load. Therefore, we put forward some suggestions for the vibration control and dynamic design of practical spatial membrane structures:(i)
In the preliminary design phase, the membrane structure with high arch-to-span ratio should be adopted as far as possible for a strong resistance to external load and vibration(ii)
After the arch-to-span ratio of membrane structure was determined, increasing the pretension helps resist external load and vibration control



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*Source: 9234832-2019-07-24.xml*