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Nonlinear dynamic responses of an axially moving laminated beam subjected to both blast and thermal loads
Accepted Manuscript Nonlinear dynamic responses of an axially moving laminated beam subjected to both blast and thermal loads Y.H. Li, L. Wang, E.C. Yang
PII: DOI: Reference:
S0020-7462(17)30483-3 https://doi.org/10.1016/j.ijnonlinmec.2018.02.007 NLM 2977
To appear in:
International Journal of Non-Linear Mechanics
Received date : 1 July 2017 Revised date : 7 February 2018 Accepted date : 20 February 2018 Please cite this article as: Y.H. Li, L. Wang, E.C. Yang, Nonlinear dynamic responses of an axially moving laminated beam subjected to both blast and thermal loads, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.02.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Nonlinear dynamic responses of an axially moving laminated beam subjected to both blast and thermal loads Y.H. Li1∗, L. Wang1∗, E.C. Yang2∗ 1 2
School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, PR China School of Mechanical of Engineering, Chongqing University of Technology, Chongqing 400054, China
Abstract The nonlinear dynamic responses of an axially moving laminated beam subjected to a blast load in thermal environment is studied considering large-displacement. Firstly, the nonlinear dynamic equilibrium equation is established based on the large-displacement theory and the constitutive relation of the single layer material in thermal environment. Based on the Galerkin method, a set of ordinary differential equations is obtained. Secondly, the multiple scales method is adopted to get the nonlinear free vibration frequency. Then, the stability region of the axial velocity and temperature is derived and the truncation order is approximated by the convergence calculation of the natural frequencies. Finally, numerical calculations are performed to discuss the effects of different kinds of blast loads, axial velocity and temperature on the nonlinear dynamic responses adopting the Runge-Kutta technique. Keywords: thermal environment; blast load; axially moving; laminated beam
1. Introduction Laminated structures are generally used in aircraft, space station, car and submarine. The laminated beam is one of them, and it is sometimes in an axially moving condition, such as paper sheets, band-saws, pipes conveying fluid, robot arms, automobiles and aerospace structures. Because the working environment of these structures is different, it is sometimes affected by thermal environment and the blast load. Therefore, it is significance to study the dynamic behavior of such a structure. The vibrations of axially moving systems have been investigated for many years and still of great concern today. The first studies on the dynamics of axially moving continua system can be dated back to the year of 1950. Initially, they are mainly concerned with the vibrations of ∗
Corresponding author. Tel.: +86 028 87600793; E-mail addresses: [email protected](Y.H. Li), [email protected](L. Wang), [email protected](E.C. Yang) Preprint submitted to Elsevier
February 7, 2018
oil pipelines. Ashley and Haviland[1] developed the linear differential equations of transverse vibration of piping systems filled with liquid. Ames et al.[2] studied the forced vibration problem of a moving threadline with geometric and material nonlinearities at the same time, and gave the numerical solutions and experimental results of that system. With the increasing application of oil pipelines, the interest in axially moving structure was motivated. So far in the literature, many researchers have done the research of the axial motion system, including linear and nonlinear vibration, free and forced vibration. Wickert and Mote [3] investigated the free and forced linear transverse vibrations of an axially moving beam with the eigenfunction method. After that, they obtain the dynamic response of an axially moving string loaded with a traveling load using the Green function method. During this process, the Volterra equations with delay were derived and solved, and the experimental results were in good agreement with the numerical analysis [4]. Moverover, Wickert [5] studied the nonlinear vibration and bifurcation of an axially moving beam and formed a complete framework of that. Vestroni [6] carried out the nonlinear dynamic behavior of a simply supported axially moving beam by the Galerkin method. On the basis of the previous researches, Pellicano and Vestroni [7] investigated the complex nonlinear dynamic behavior of a high-speed axially moving beam by means of a high-dimensional discrete model. Chakraborty and Mallik [8] addressed the free and forced vibration of a nonlinear axially moving beam which has an intermediate guide. Ghayesh and Balar [9] analyzed the parametric vibration and stability of an axially moving viscoelastic Rayleigh beam considering the geometric nonlinearity. Later, Ghayesh et al. [10] did the similar works in an axially moving laminated composite beam with the constant axial velocity. Huang [11] focused on the nonlinear vibration and bifurcation of an axially moving beam, which is subjected to a periodic lateral force excitation. The previous researches were mainly based on uniform axial moving, but there are also many ¨ et al. studies dealt with the axially moving beam with the time-dependent axial velocity. Oz [12] investigated the nonlinear vibration of an axially moving beam with the method of multiple scales, where the beam is moving with a time-dependent velocity. The stability of the structure ¨ under three different velocity functions are discussed. Ozhan [13] aimed to analyze the stability of an axially moving damped beam under the combined effects of variable speed and axial force. For the axially accelerating viscoelastic beam, Chen and Yang did the contribution. Chen et al. [14] analyzed the dynamic stability of a viscoelastic beam subjected to an axially accelerated speed. Later, considering the geometric nonlinearity, Yang et al. [15] re-derived the bifurcation and chaos of the accelerating viscoelastic beam. Moreover, for the effect of blast load on the vibration behaviors of the beam without the axial velocity, Zhai et al. [16] performed the characteristics of a reinforced concrete beam subjected to a blast load using the numerical and experimental method, respectively. Besides, there are a lot of researchers interested in the effect of blast load on reinforced concrete beam [17–20]. Nassr et al. [21] studied strength and stability of a steel beam column under blast load. Chernin et al. [22] presented an analytical method to predict the blast dynamic responses of a beam-column subjected to the combined effects of axial and transverse loads. Langdon [23] analyzed the dynamic responses of a composite sandwich structure with air-blast load using the ˇ experimental and numerical means. Sliseris et al.[24] established a nonlinear model to calculate 2
the dynamic behavior of cross laminated timber under the blast load. The numerical model was compared with classical beam theory and the results show that the shear deformation must be taken into account when designing CLT. Bodaghi et al. [25] studied the nonlinear active control of a functionally graded beam subjected to a blast load in thermal environment. Obviously, much research has been done about the effect of axial velocity on dynamics of an axially moving structure, and the influence of blast load on the beam without axial moving has been also studied by many researchers. However, the analysis for considering both effects of blast load and temperature or the combined effects of axial velocity, blast load, and temperature, are still far from perfect. To address the lack of that, we establish a mathematical model to investigate the nonlinear dynamic behavior of a symmetrically laminated beam subjected to the combined effects of blast load, axially moving and thermal environment. The rest of this paper is organized as follows. In Section 2, the second-order ordinary differential governing equation of a symmetrically laminated beam is established, and the beam with axial velocity is subjected to a blast load in thermal environment. In Section 3, the frequency of the nonlinear free vibration is obtained by using the multiple scale method. In Section 4, the relationship of the temperature variation and the axial velocity in critical condition is derived. In Section 5, the effects of different kinds of blast loads, axial velocity and temperature variation are discussed regarding to the nonlinear responses of the system. 2. Governing equations 2.1. Model of laminated beam Consider an axially moving laminated beam subjected to a blast load in thermal environment. The model of the beam is depicted in Fig. 1 and the cross section is shown in Fig. 2, in which l is the length, b is the width and H is the total thickness of the beam, vx is the axial velocity of the beam in x-direction, w(x, t) is the lateral deflection and P (t) is the blast load. The total number of the layers is n and the fibre orientation lies symmetrically with respect to the mid-plane of the beam. The thickness of the k − th layer is tk = zk − zk−1 .
Fig. 1. Model of the laminated beam
2.2. Basic equations
3
Fig. 2. Cross section model of the laminated beam
On the basis of the Euler-Bernoulli beam theory, the displacement field of the beam can be expressed as u1 = u − zw,x u2 = 0 u3 = w
(1)
where ui (i = 1, 2, 3) represent the displacements in x-, y- and z-direction at any point. u and w are axial and transverse displacements of the mid-plane. Employing the large-displacement assumption, the the axial strain of the beam can be obtained as εxx = ε0xx − zw,xx
(2)
where ε0xx represents the mid-plane strain and given as [26] 1 2 ε0xx = u,x + w,x 2
(3)
Considering the thermal effect of the composite material, the axial stress-strain relation of the present laminated beam model can be expressed as σxx = Q11 [εxx − αx ∆T ]
(4)
where αx = α1 cos2 θ + α2 sin2 θ
(5)
4
and Q11 = Q11 cos4 θ + 2(Q12 + 2Q66 ) sin2 θ cos2 θ + Q22 sin4 θ
(6)
where α1 and α2 are the thermal expansion coefficients of a single layer material along the fibre orientation and perpendicular to the fiber direction, respectively. ∆T is the variation of the temperature. θ is the fiber orientation angle and it is defined as the angle from x-axis to the fiber normal direction. Qij (i, j = 1, 2, 6) are elements of the stiffness of single layer material and have the following forms E1 E2 , Q22 = Q11 = (1 − ν12 ν21 ) (1 − ν12 ν21 ) E1 ν12 Q12 = , Q66 = G12 (1 − ν12 ν21 ) where E1 and E2 are elastic moduli of the material in principle directions and G12 is shear modulus, ν12 and ν21 are Poisson’s ratios.
2.3. Derivation of transverse governing equation The governing equation of motion can be obtained using the Hamilton’s principle, which can be expressed as ∫ t2 (δU − δV − δK)dt = 0 (7) t1
where δ is the variation operator, K is the kinetic energy, U is the potential energy, V is the work done by external force P (t). The variation of kinetic energy is written as δK =
∫ l∫ 0
ρ(uδ ˙ u˙ + wδ ˙ w)dAdx ˙
(8)
A
and the the variation of strain energy and the variation of work are defined as ∫ l∫ δU = σij δϵij dAdx 0
δV =
∫
(9)
A
l
P (t)δwdx
(10)
0
where ρ is the material density and A is the cross-sectional area of the beam. Then, we define the axial resultant Nxx and the bending moment Mxx in unit width as follows
5
[ ] Nxx , Mxx =
∫
A
[ ] σxx 1, z dA
(11)
And then Eqs. (8) and (9) can be rewritten as δK =
δU =
∫ ∫
l
ρA(uδ ˙ u˙ + wδ ˙ w)dx ˙
(12)
[Nxx (δu,x + w,x δw,x ) − Mxx δw,xx ]dx
(13)
0
0
l
Substituting Eqs. (10), (12), and (13) into Eq. (7), the governing equations with respect to axial resultant and bending moment can be obtained Nxx,x = ρA¨ u
(14)
Nxx,x w,x + Nxx w,xx + Mxx,xx + P (t) = ρAw¨
(15)
On the basis of Ref. [6], the term ρAw¨ of the axially moving beam can be given as ρAw¨ = ρA(
2 ∂ 2w ∂ 2w 2∂ w + 2v + v ) x x ∂t2 ∂x∂t ∂x2
(16)
Combining Eqs. (2), (4) and (11), one can derive 1 2 Nxx = A11 (u,x + w,x ) − B11 w,xx − NxT 2 1 2 Mxx = B11 (u,x + w,x ) − D11 w,xx − MxT 2
(17)
where A11 , B11 and D11 are the tensile, stretching-bending coupling and bending stiffness, respectively. They are denoted as [ ] A11 , B11 , D11 =
∫
H/2
−H/2
[ ] Q11 1, z, z 2 dz
(18)
NxT and MxT are the additional membranous force and the additional bending moment generated
6
by the temperature variation. They are defined as [ T ] Nx , MxT =
∫
H/2
−H/2
[ ] Q11 1, z αx ∆T dz
(19)
Assuming the in-plane axial inertia force is to zero, the axial resultant in Eq. (14) can be expressed as a function independent of x. Then integrating Eq. (17) into Eq. (14) and solving u yields 1 u=− 2
∫
x
0
w,ξ2 dξ +
1 T 1 B11 w,x + Nx + b1 (t)x + b2 (t) A11 A11 A11
(20)
where b1 (t) and b2 (t) are the integral undetermined term. The axial displacement boundary conditions are shown as u|x=0 = 0 u|x=L = 0
(21)
Substituting Eq.(20) into Eq. (21), the b1 (t) and b2 (t) can be obtained ∫ A11 l 2 B11 b1 (t) = w,x dx − [w,x (l) − w,x (0)] − NxT 2l 0 l B11 b2 (t) = − w,x (0) A11
(22)
with the help of Eqs. (20) and (22), the Eq. (15) can be rewritten as ∫ 2 B11 A11 l 2 T (D11 − )w,xxxx + [Nx − w dx]w,xx A11 2l 0 ,x ∂2w ∂ 2w ∂ 2w = P (t) − ρA( 2 + 2vx + vx2 2 ) ∂t ∂x∂t ∂x
(23)
2.4. Galerkin discretization The governing equation given by Eq. (23) can be reduced in time domain by choosing the approximation function for displacement field and employing the Galerkin method. The approximate solution is usually represent as [6]. w(x, t) =
M ∑
Um (t)ϕm (x)
(24)
m=1
where ϕm (x) is the trial function, M is the truncation order, and the choice of the trial function depends on the boundary condition. The trial functions for different boundary conditions are 7
shown as Hinged-Hinged(HH): w|x=0,l = 0; w,xx |x=0,l = 0 ϕm (x) = sin(
(25)
mπx ) l
Clamped-Clamped(CC): w|x=0,l = 0; w,x |x=0,l = 0 ϕm (x) = cos(km x) − cosh(km x) + [
sin(km l) + sinh(km l) ][sin(km x) − sinh(km x)] cos(km l) − cosh(km l)
(26)
Frequency equation(CC): cos(km l) · cosh(km l) = 1 Clamped-Free(CF): w|x=0 = 0; w,x |x=0 = 0 ϕm (x) = cos(km x) − cosh(km x) − [
cos(km l) + cosh(km l) ][sin(km x) − sinh(km x)] sin(km l) − sinh(km l)
(27)
Frequency equation(CF): cos(km l) · cosh(km l) = −1 For the general result, introducing Eq. (24) into Eq. (23) and employing the Galerkin principle, a set of ordinary differential equations can be obtained M ∑
mmi U¨i +
i=1
M ∑ i=1
cmi U˙ i +
M ∑ i=1
kmi Ui −
M ∑ M ∑ M ∑
nl kmikl Ui Uk Ul = pm
(28)
i=1 k=1 l=1
where mmi = ρHb
∫
l
ϕi (x)ϕm (x)dx ∫ l 2 ∫ l B11 ′′′′ T 2 kmi = (D11 − ) ϕ (x)ϕm (x)dx + (Nx + ρHbvx ) ϕ′′i (x)ϕm (x)dx A11 0 i 0 ∫ l ϕ′i (x)ϕm (x)dx cmi = 2ρHbvx 0 ∫ l pm = P (t)ϕm (x)dx 0 ∫ ∫ l A11 l ′′ nl ϕm (x)ϕi [ ϕ′k (x)ϕ′l (x)dx]dx kmikl = 2l 0 0 0
8
(29)
3. Nonlinear free vibration The multiple scales method is adopted to solve the nonlinear free vibration equation that is Eq. (28) without the external force, and it can be rewritten as follows U¨j + Cj U˙ j + ωj2 Uj −
M ∑ M ∑ M ∑
nl gjmkl Um Uk Ul = 0
(30)
m=1 k=1 l=1
where ωj is the linear frequency that can be obtained from the mass matrix and stiffness matrix. In the light of nonlinear dynamics theories in Ref. [27], the expression of Uj , its derivatives ˙ Uj and U¨j are given as Uj =εUj1 (T0 , T1 , T2 ) + ε2 Uj2 (T0 , T1 , T2 ) + ε3 Uj3 (T0 , T1 , T2 ) + ... U˙j =ε(D0 Uj1 ) + ε2 (D0 Uj2 + D1 Uj1 ) + ε3 (D0 Uj3 + D1 Uj2 + D2 Uj1 ) + ... U¨j =εD2 Uj1 + ε2 (D2 Uj2 + 2D0 D1 Uj1 )+ 0
(31)
0
ε3 Uj3 (D02 + 2D0 D1 Uj2 + D12 Uj1 + 2D0 D2 Uj1 ) + ... where Di = ∂/∂Ti , the small parameter ε is a book-keeping device, T0 , T1 and T2 are different time scales which can be expressed as t, εt and ε2 t, respectively. Substituting Eq. (31) into Eq. (30) collecting terms of the same order of ε yields Order ε1 : D02 Uj1 + ωj2 Uj1 = 0 f or
j = 1, 2, ...
(32)
Order ε2 : D02 Uj2 + ωj2 Uj2 = −2D0 D1 Uj1
f or
j = 1, 2, ...
(33)
Order ε3 : D02 Uj3 + ωj2 Uj3 = −
D12 Uj1
− 2D0 D2 Uj1 − 2D0 D1 Uj1 +
M ∑ M ∑ M ∑
nl gjmkl Um Uk Ul
f or
j = 1, 2, ...
(34)
m=1 k=1 l=1
Assuming the general solution of Eq. (32) is [27] Uj1 = Aj (T1 , T2 )eiωj T0 + CC
(35)
where Aj (T1 , T2 ) is a function to be determined, CC is the complex conjugate of the preceding term. Substituting Eq. (35) into Eq. (33) yields D02 Uj2 + ωj2 Uj2 = −2iωj D1 Aj (T1 , T2 )eiωj T0 + CC 9
(36)
Eliminating the secular term of Eq. (36), we have −2iωj D1 Aj (T1 , T2 ) = 0
(37)
Thus it can be seen, Aj (T1 , T2 ) is only a function of T2 . Inserting Eq. (37) into Eq. (36) derives Uj2 = 0
(38)
Substituting Eq. (35), Eq. (36) and Eq. (37) into Eq. (34), the following equation can be given by D02 Uj3 + ωj2 Uj3 = − 2iωj D1 A′j (T2 )eiωj T0 +
nl gjjjj [A3j (T2 )e3iωj T0 + 3A2j (T2 )Aj (T2 )eiωj T0 ] + cc
(39)
Similarly, eliminating the secular term of Eq. (39), we have nl −2iωj A′j (T2 ) + 3gjjjj A2j (T2 )Aj (T2 ) = 0
(40)
Letting Aj = (1/2)αj (T2 )eiβj (T2 ) , and inserting it into Eq. (40), two differential equations can be obtained as d αj (T2 ) = 0 dT2
(41)
d 3 nl 2 βj (T2 ) = − gjjjj a dT2 8
(42)
The solutions of Eqs. (41)-(42) are as follow αj (T2 ) = αj0 ,
(43)
3 nl 2 βj (T2 ) = − gjjjj a T2 + βj0 8
(44)
where aj0 and βj0 are constants, which depend on the initial conditions. Substituting Eqs. (43)(44) into the expression of Aj , and inserting the result into Eq. (35), the nonlinear frequency can be denoted ωnj = ωj −
3 nl 2 g a . 8ωj jjjj j0
(45)
10
4. The stability region In order to provide the theoretical basis for subsequent analysis, the relationship of the axial velocity and the temperature variation in critical condition must be determined first. For a simply supported symmetrically laminated beam, the trial function ϕm (x) is given by Eq.(25). Taking the truncation term M = 1 and substituting Eq. (25) into Eq. (28) of which the nonlinear term and the external force is not included, thus the system equation can be expressed as M1 U¨1 + C1 U˙1 + K1 U1 = 0
(46)
where ∫
l ρAl , M1 = ρHb ϕ21 (x)dx = 2 0 ∫ l ∫ l (4) (2) T 2 K1 = D11 ϕ1 (x)ϕ1 (x)dx + (Nx + ρHbvx ) ϕ1 (x)ϕ1 (x)dx 0 4
0
2
π D11 π − (NxT + ρAvx2 ) , = 3 2l 2l ∫ l C1 = 2ρHbvx ϕ′1 (x)ϕ1 (x)dx = 0.
(47)
0
Then we arrive at ω12
D11 π 4 π2 T 2 = K1 /M1 = − (Nx + ρAvx ) ρAl4 ρAl2
(48)
It satisfies the following inequality D11 π 4 π2 T 2 − (N + ρAv ) ≥0 x x ρAl4 ρAl2
(49)
Hence, we can derive C∆T +
ρAvx2
D11 π 2 ≤ ρl2
(50)
where C=
n ∑ k=1
(Q11 αx + Q12 αy + Q16 αxy )k (zk − zk−1 )
11
(51)
The stability region R is defined as } { 2 D π 11 R = (∆T, vx ) |C∆T + ρAvx2 ≤ ρl2
(52)
For arbitrary values of the axial velocity and the temperature variation belonged to the region D11 π 2 2 is the critical (∆T, vx ) ∈ R, the motion is stable. Physically speaking, C∆T + ρAvx = ρl2 condition for instability. Similarly, taking the truncation term M = 2 and following the step described above, we obtained ¨ + C∗ U ˙ + K∗ U = 0 M∗ U
(53)
where
ρAl 0 [m 0 ] 2 ∗ M = = ρAl 0 m 0 2
(54)
(55)
D11 π 4 π2 [ ] T 2 0 k11 0 2l3 − (Nx + ρAvx ) 2l ∗ K = = 8D11 π 4 0 k22 π2 T 2 0 − 2(N + ρAv ) x x l3 l
C∗ =
0 8 ρAvx 3
8 − ρAvx [0 −c] 3 = c 0 0
(56)
The characteristic equation of the system is [27] 2 ∗ λ M + λC∗ + K∗ = 0
Solving the Eq. (57), we can readily derive √ √ −B− A λ1,2 = ± i 2m2 √ √ −B+ A i λ3,4 = ± 2m2
(57)
(58)
12
where 2
A = B − 4k11 k22 m2
B = k11 m + k22 m + c2 According to Eq. (58), we can arrive at {
} 2 D π 11 R1 = (∆T, vx ) |C∆T + ρAvx2 ≤ l2 { } 2 4D π 11 R2 = (∆T, vx ) |C∆T + ρAvx2 ≤ l2
(59)
where R1 and R2 are the stability regions of the fundamental frequency and the high frequency, respectively. As we can see, the region R1 is a subset of the region R2 . On the analogy of this, we can find that the low frequency stability region is a subset of the hight frequency stability region as the value of the truncation term M is greater. For the other various boundary conditions the corresponding stable domain can be obtained by similar methods. 5. Numerical results In this section, the numerical results for the symmetrically laminated beam subjected to a blast load in thermal environment are performed by using the fourth-order Runge-Kutta method. However, considering the influence of the axial velocity, the one-term approximation formula for the displacement function is not applicable to this study. Therefore, how to determine the truncation order M is also a mission in this paper. The elastic modulus of graphite/epoxy with various temperature is given in Table 1 [28] and the geometric parameter in Table 2 are taken as an example in the following analysis. The density of the beam is ρ = 1600kg/m3 , the ply angle is (0◦ /90◦ )s . The thermal expansion coefficients and Poisson’s ratio are taken as α1 = −0.3 × 10−6 , α2 = 28.1 × 10−6 , and v21 = 0.3. Table 1 Elastic moduli of graphite/epoxy composite.
Moduli(GPa)
Temperature, T (◦ C) 25
50
75
100
125
150
E1
130
130
130
130
130
130
E2
9.5
9.0
8.5
8.0
7.5
6.75
G12
6.0
6.0
5.5
5.0
4.75
4.5
In Section 2, ∆T = T − T0 is defined as temperature variation. To unify the discussion, we define T0 = 25◦ C as referenced values in the present study. 13
Table 2 Geometric parameters of the laminate beam
Length(m)
Width(m)
Thickness(m)
Layers
1
0.05
0.0125
4
5.1. The critical curve Based on Eq. (50) and Eq. (59), the one-two orders critical curves about the axial velocity vx and the temperature variation ∆T can be obtained, which are shown in Fig. 3 and Fig. 4.
Fig. 3. One orders critical curve
Fig. 4. Two orders critical curve
As indicated in Fig. 3 and Fig. 4, the lines illustrate the relationship between the temperature variation ∆T and the axial velocity vx in the critical state. In other words, the area below the line is the stability region as defined in Section 4. Noticeably, the vibration of the laminated beam is stable as the values of ∆T and vx are in this area. Comparing Fig. 3 and Fig. 4 shows that the stability region of the high-order frequency is larger than that of the low-order. Generally speaking, in order to meet the requirement, the values of ∆T and vx should be in the stability region of the low-order. This phenomenon is similar to the result in Section 4. 5.2. Convergence validation To determine the truncation order M , the frequency of the laminated beam under different boundary conditions with different M are performed. The blast load is an uniform pressure, which is selected the explosive blast load as an example in this subsection, and it is given by [29] { Pm (1 − t/tp )e−ηt/tp P (t) = 0
0 ≤ t ≤ tp t > tp
(60)
14
where Pm is the peak blast pressure, tp is the positive phase duration, η is the waveform parameter. For the next calculation, the parameters in Table 3 are adopted. Table 3 Relevant parameters for calculation
Pm (MPa) 3.447
tp (s) 0.02
η 2
vx (m/s) 50
∆T (◦ C) 30
The influence of the truncation order M on the frequency is discussed under different boundary conditions by using the eigenvalue method and it is shown in Fig. 5. The HH, CC, and CF represent the boundary conditions as mentioned in section 2.4.
(a)
(b)
(c)
Fig. 5. Variation of frequency with truncation order M under different boundary conditions
It can be found from Fig. 5 that the frequencies of the three boundary conditions (HH, CC, and CF) achieve convergence as the M takes 10. Hence the truncation order M can be considered as 10 for the following analysis. 5.3. Comparison validation In order to investigate the accuracy of the above prediction, a finite element simulation model is carried out employing LS-DYNA software. The hexahedral elements with 20 nodes are used in the modeling process and the total number of elements is 1000. The response at the centre of the simply supported laminated beam is solved by using the Runge-Kutta method developed in this paper, in which the parameters of axial velocity, positive phase duration, maximum blast pressure, and temperature variation are the same with the previous case. The result is compared with that obtained by the finite element method (FEM), and the comparison is presented in Fig. 6. It indicates that the two sets of results are in good agreement with each other. 5.4. Results and discussions Firstly, for the simply supported symmetrically laminated beam, the effect of the axial velocity on the variation of the response is detailed in Fig. 7. It can be observed that the variation of the response is imperceptible in time domain [0,0.01]s and obvious in [0.01,0.02]s. In order 15
Fig. 6. Responses of the present and FEM
to facilitate observation, Fig. 8 gives the variation of the response versus time in time domain [0.01,0.02]s particularly.
Fig. 7. Effect of the axial velocity on the flexural amplitude
It is evident from the result that the flexural amplitude decreases while the axial velocity increases due to the gyro-effect. The reasons are as follows: from Eq. (23), the axial velocity affects the stiffness and damping of the structure, but we don’t know which one has a greater impact on the response. Therefore, the following discussions are made. Letting Kk and Cc instead of the terms ρAvx2 and ρAvx in Eq. (23) yields 3 2 − NxT − Kk )w,xx = −P (t) −2Cc w,xt − ρAw,tt − D11 w,xxxx + ( A11 w,x 2
(61)
Based on this equation, four cases are considered, including: (1) Kk = 0, Cc = 0, this case is not taken into account the axial velocity; (2) Cc = 0, Kk ̸= 0, which means the axial velocity only affects the stiffness; (3) Kk = 0, Cc ̸= 0, it represents the axial velocity only affects the damping; (4) Kk ̸= 0, Cc ̸= 0, it is the model studied in this paper. The amplitude responses of the four cases are shown in Fig. 9. As you can see in Fig. 9, the amplitude response of the second case is more nearly the model studied in this paper, which means the influence of 16
Fig. 8. Variation of flexural amplitude versus time in the range of 0.01s to 0.02s
damping on response is greater than that of stiffness. Therefore, the flexural amplitude decreases while the axial velocity increases due to the gyro-effect, and the greater the velocity, the greater the damping. In addition to, as the the axial velocity increases, the period of vibration becomes longer.
Fig. 9. Variation of flexural amplitude for the four cases versus time
On the other hand, the motion state of the system changes into nonlinear free vibration while t > ηtp as mentioned by Eq. (60), and it is listed in Fig. 10. The datum in Fig. 10 suggest that the nonlinear response and frequency of free vibration are related to the initial condition and linear frequency, which is agreement with the Eq. (45). Secondly, the time history of the amplitude at the centre of the laminated beam with different temperatures is shown in Fig. 11. It explains that the vibration period and amplitude of the laminated beam increases with the increase of the temperature. As we know, the temperature has an influence on the stiffness of the structure, and the higher the temperature variation, the smaller the stiffness will be. In the main, the smaller the stiffness, the lager the flexural amplitude. So we can found from Fig. 11 that with increasing the temperature, the flexural 17
Fig. 10. Effect of the axial velocity on the nonlinear free vibration
amplitude increases, and the frequency decreases.
Fig. 11. Effect of the temperature on the flexural amplitude
Fig. 12 gives the response at the centre of the laminated beam versus time with different positive phase durations. As evident from Fig. 12, the increase of the positive phase duration tp leads to the increase of the amplitude. Physically speaking, as the tp increasing, the blast load subjected to the beam will be lasted for a longer time so that the amplitude will be increased. Furthermore, the period of vibration is found to be shorten with increasing positive phase duration. Fig. 13 illustrates the influence of different peak blast pressures on the response at the centre of the laminated beam. As can be seen from the approximate-numerical result in Fig. 13, the flexural amplitude and frequency increase while the peak blast pressure increases. Then, a comparison of the deflection time history of the laminated beam with different ply angles is shown in Fig. 14. It indicates that as the ply angle is (0◦ /0◦ )s , the flexural amplitude is the minimum. On the contrary, when the ply angle is (90◦ / − 90◦ )s , the flexural amplitude reaches the maximum. In addition, the vibration response of the laminated beam with different boundary conditions under explosive blast load is shown in Fig. 15. As we can see from the Fig. 15, when the boundary condition is CF, the amplitude and 18
Fig. 12. Effect of the positive phase duration on the flexural amplitude
Fig. 13. Effect of the peak blast pressure on the flexural amplitude
Fig. 14. Effect of the ply angle on the flexural amplitude
19
Fig. 15. Effect of the boundary condition on the flexural amplitude
the period of the vibration are the biggest, while the amplitude and the period are the smallest when the boundary condition is CC. Furthermore, there are various types of blast loads. In the following analysis, the effects of different blast loads on the responses of the laminated beam are discussed. Firstly, the triangular load is given by [29] { Pm (1 − t/tp ) 0 ≤ t ≤ rtp P (t) = 0 t < 0 and t > rtp
(62)
where r indicates the shock pulse length factor. The shape of the pulse is decided by r. For the case of r = 1, it is called air-blast, for r = 2, it is antisymmetric about x-axis and called sonic boom. Substituting Eq. (62) into Eq. (28) and solving it, the results for the case of r = 1 with different axial velocities and temperatures are shown in Fig. 16 and Fig. 17, respectively.
Fig. 16. Effect of the axial velocity on the response of the beam subjected to the air-blast
As can be observed in Fig. 16 and Fig. 17, there is a slight changes of the flexural amplitude with the increasing axial velocity and temperature, which means that a single factor, the axial velocity or the temperature has relatively limited influence on the flexural amplitude and the 20
Fig. 17. Effect of the temperature on the response of the beam subjected to the air-blast
vibration frequency. What’s more, it also shows that the air-blast has a greater impact on the structure than the explosive blast load. Secondly, the numerical result for laminated beam subjected to the nuclear blast load is performed. The function of the pulse is given by [29] { Pm (1 − t/tp ) 0 ≤ t ≤ t1 P (t) = P0 (1 − t/t2 ) t1 ≤ t ≤ t2
(63)
This pulse is consist of two linear functions, the Pm is the same with the above and we choose the P0 = 0.5Pm , tp = 0.01s, and t2 = 0.02s for the function as an example, and the t1 is (Pm − P0 )(tp t2 )/(Pm t2 − P0 tp ). For the reason that the pulse is in accordance with Eq. (62) as the time in the range of 0 < t < t1 , hence we mainly focus on the response for the second part of the pulse and its results are presented in Fig. 18 and Fig. 19. It can be identified that the amplitude decreases/increase while the axial velocity/temperature increase.
Fig. 18. Effect of the axial velocity on the response of the beam subjected to the nuclear pulse
In the end, the dynamic response of the laminated beam subjected to the sine pulse is 21
Fig. 19. Effect of the temperature on the response of the beam subjected to the nuclear pulse
analyzed, and the sine pulse can be expressed as [29] { Pm sin(πt/rtp ) 0 ≤ t ≤ rtp P (t) = 0 t < 0 and t > rtp
(64)
The pulse is a periodic excitation, therefore, in this case, the amplitude and the vibration frequency of the beam mainly depend on the rtp of the periodic excitation. Based on this, the axial velocity and the temperature hardly have any influence on the flexural amplitude and the vibration frequency, as seen in Fig. 20 and Fig. 21.
Fig. 20. Effect of the axial velocity on the response of the beam subjected to the sine pulse
6. conclusion In this work, the nonlinear dynamic response of a symmetrically laminated beam is studied under the framework of geometric nonlinearity, and the combined effects of the axial velocity, blast load and thermal environment are considered. The nonlinear coupled equation in the time domain is obtained by employing the Galerkin method. The truncation order of the approximate 22
Fig. 21. Effect of the temperature on the response of the beam subjected to the sine pulse
solution is selected by convergence calculation of the dynamic response solved by the RungeKutta method. Through numerical analyses, the effects of some detailed parameters of the laminated beam on the nonlinear dynamic response are discussed. The results of this work can be summarized as follow: 1) For nonperiodic blast load: a) Due to the gyro-effect, the flexural amplitude and frequency decrease while the axial velocity increases. b) With increasing temperature, the flexural amplitude increases, and the frequency decreases. 2) For the explosive blast load: a) As the positive phase duration tp increases, the flexural amplitude and frequency tend to increase. b) The flexural amplitude and frequency increase while the increase of the peak reflected pressure. c) As the ply angle is (0◦ /0◦ )s , the flexural amplitude is the minimum, when the ply angle is (90◦ / − 90◦ )s , the flexural amplitude reaches the maximum. 3) For the sine pulse, the amplitude and vibration frequency of the beam mainly depend on the periodic excitation. Namely, the axial velocity and the temperature hardly have any influence on the flexural amplitude and vibration frequency. 4)When the boundary condition is CF, the amplitude and the period of the vibration are the biggest, while the amplitude and the period are the smallest when the boundary condition is CC. Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 11372257, 11472064, 11602208, 51674216), the Science and Technology Foundation of ChongQing Education Commission(Nos. KJ1500929), the project of Chongqing University of Science and Technology (Nos. Shljzyh2017-007). The supports from the innovative cultivation fund of the excellent doctoral graduate students of Southwest Jiao Tong University are gratified as well. References [1] H Ashley and G Haviland. Bending vibrations of a pipe line containing flowing fluid. Journal of Applied Mechanics-Transactions of the ASME, 17(3):229–232, 1950.
23
[2] WF Ames, SY Lee, and JN Zaiser. Non-linear vibration of a traveling threadline. International Journal of Non-linear mechanics, 3(4):449–456, 1968. [3] JA Wickert and CD Mote Jr. Classical vibration analysis of axially moving continua. Journal of Applied Mechanics of the ASME, 57(3):738–744, 1990. [4] Mote Jr CD Wickert JA. Travelling load response of an axially moving string. Journal of Sound and Vibration, 149(2):267–284, 1991. [5] JA Wickert. Non-linear vibration of a traveling tensioned beam. International Journal of Non-Linear Mechanics, 27(3):503–517, 1992. [6] F Vestroni. Nonlinear dynamics and bifurcations of an axially moving beam. Journal of Vibration and Acoustics, 122(1):21–30, 2000. [7] F Pellicano and F Vestroni. Complex dynamics of high-speed axially moving systems. Journal of Sound and Vibration, 258(1):31–44, 2002. [8] G Chakraborty and AK Mallik. Non-linear vibration of a travelling beam having an intermediate guide. Nonlinear Dynamics, 20(3):247–265, 1999. [9] MH Ghayesh and S Balar. Non-linear parametric vibration and stability of axially moving visco-elastic rayleigh beams. International Journal of Solids and Structures, 45(25):6451–6467, 2008. [10] MH Ghayesh, M Yourdkhani, S Balar, and T Reid. Vibrations and stability of axially traveling laminated beams. Applied Mathematics and Computation, 217(2):545–556, 2010. [11] JL Huang, RKL Su, WH Li, and SH Chen. Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. Journal of Sound and Vibration, 330(3):471–485, 2011. ¨ M Pakdemirli, and H Boyacı. Non-linear vibrations and stability of an axially moving beam with [12] HR Oz, time-dependent velocity. International Journal of Non-Linear Mechanics, 36(1):107–115, 2001. ¨ [13] B B Ozhan. Vibration and stability analysis of axially moving beams with variable speed and axial force. International Journal of Structural Stability and Dynamics, 14(06):1450015, 2014. [14] LQ Chen, XD Yang, and CJ Cheng. Dynamic stability of an axially accelerating viscoelastic beam. European Journal of Mechanics-A/Solids, 23(4):659–666, 2004. [15] XD Yang and LQ Chen. Bifurcation and chaos of an axially accelerating viscoelastic beam. Chaos, Solitons & Fractals, 23(1):249–258, 2005. [16] CC Zhai, L Chen, HB Xiang, and Q Fang. Experimental and numerical investigation into rc beams subjected to blast after exposure to fire. International Journal of Impact Engineering, 97:29–45, 2016. [17] Wensu Chen, Hong Hao, and Shuyang Chen. Numerical analysis of prestressed reinforced concrete beam subjected to blast loading. Materials & Design (1980-2015), 65:662–674, 2015. [18] YS Tai, TL Chu, HT Hu, and JY Wu. Dynamic response of a reinforced concrete slab subjected to air blast load. Theoretical and applied fracture mechanics, 56(3):140–147, 2011. [19] G Carta and F Stochino. Theoretical models to predict the flexural failure of reinforced concrete beams under blast loads. Engineering structures, 49:306–315, 2013. [20] Flavio Stochino. Rc beams under blast load: Reliability and sensitivity analysis. Engineering Failure Analysis, 66:544–565, 2016. [21] Amr A Nassr, A Ghani Razaqpur, Michael J Tait, Manuel Campidelli, and Simon Foo. Strength and stability of steel beam columns under blast load. International Journal of Impact Engineering, 55:34–48, 2013. [22] L Chernin, M Vilnay, and I Shufrin. Blast dynamics of beam-columns via analytical approach. International Journal of Mechanical Sciences, 106:331–345, 2016. [23] GS Langdon, CJ Von Klemperer, BK Rowland, and GN Nurick. The response of sandwich structures with composite face sheets and polymer foam cores to air-blast loading: preliminary experiments. Engineering Structures, 36:104–112, 2012. ˇ [24] J Sliseris, L Gaile, and L Pakrasti¸nˇs. Numerical analysis of behaviour of cross laminated timber (clt) in blast loading. In IOP Conference Series: Materials Science and Engineering, volume 251, page 012105. IOP Publishing, 2017. [25] M Bodaghi, AR Damanpack, MM Aghdam, and M Shakeri. Non-linear active control of fg beams in thermal environments subjected to blast loads with integrated fgp sensor/actuator layers. Composite Structures, 94(12):3612–3623, 2012.
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[26] S Timoshenko and SW Krieger. Theory of plate and shell. McGraw-Hill of America New York, 1959. [27] R Grimshaw. Nonlinear ordinary differential equations, volume 2. CRC Press, 1991. [28] Makhecha D P Patel B P, Ganapathi M. Hygrothermal effects on the structural behaviour of thick composite laminates using higher-order theory. Composite Structures, 56(1):25–34, 2002. [29] Z Kazancı. Dynamic response of composite sandwich plates subjected to time-dependent pressure pulses. International Journal of Non-Linear Mechanics, 46(5):807–817, 2011.
25
Highlights
The geometrical nonlinearity of the structure is considered.
The axial velocity, temperature and blast load are considered at the same time.
Effects of axial velocity and temperature on the nonlinear response are discussed.
PII: DOI: Reference:
S0020-7462(17)30483-3 https://doi.org/10.1016/j.ijnonlinmec.2018.02.007 NLM 2977
To appear in:
International Journal of Non-Linear Mechanics
Received date : 1 July 2017 Revised date : 7 February 2018 Accepted date : 20 February 2018 Please cite this article as: Y.H. Li, L. Wang, E.C. Yang, Nonlinear dynamic responses of an axially moving laminated beam subjected to both blast and thermal loads, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.02.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Nonlinear dynamic responses of an axially moving laminated beam subjected to both blast and thermal loads Y.H. Li1∗, L. Wang1∗, E.C. Yang2∗ 1 2
School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, PR China School of Mechanical of Engineering, Chongqing University of Technology, Chongqing 400054, China
Abstract The nonlinear dynamic responses of an axially moving laminated beam subjected to a blast load in thermal environment is studied considering large-displacement. Firstly, the nonlinear dynamic equilibrium equation is established based on the large-displacement theory and the constitutive relation of the single layer material in thermal environment. Based on the Galerkin method, a set of ordinary differential equations is obtained. Secondly, the multiple scales method is adopted to get the nonlinear free vibration frequency. Then, the stability region of the axial velocity and temperature is derived and the truncation order is approximated by the convergence calculation of the natural frequencies. Finally, numerical calculations are performed to discuss the effects of different kinds of blast loads, axial velocity and temperature on the nonlinear dynamic responses adopting the Runge-Kutta technique. Keywords: thermal environment; blast load; axially moving; laminated beam
1. Introduction Laminated structures are generally used in aircraft, space station, car and submarine. The laminated beam is one of them, and it is sometimes in an axially moving condition, such as paper sheets, band-saws, pipes conveying fluid, robot arms, automobiles and aerospace structures. Because the working environment of these structures is different, it is sometimes affected by thermal environment and the blast load. Therefore, it is significance to study the dynamic behavior of such a structure. The vibrations of axially moving systems have been investigated for many years and still of great concern today. The first studies on the dynamics of axially moving continua system can be dated back to the year of 1950. Initially, they are mainly concerned with the vibrations of ∗
Corresponding author. Tel.: +86 028 87600793; E-mail addresses: [email protected](Y.H. Li), [email protected](L. Wang), [email protected](E.C. Yang) Preprint submitted to Elsevier
February 7, 2018
oil pipelines. Ashley and Haviland[1] developed the linear differential equations of transverse vibration of piping systems filled with liquid. Ames et al.[2] studied the forced vibration problem of a moving threadline with geometric and material nonlinearities at the same time, and gave the numerical solutions and experimental results of that system. With the increasing application of oil pipelines, the interest in axially moving structure was motivated. So far in the literature, many researchers have done the research of the axial motion system, including linear and nonlinear vibration, free and forced vibration. Wickert and Mote [3] investigated the free and forced linear transverse vibrations of an axially moving beam with the eigenfunction method. After that, they obtain the dynamic response of an axially moving string loaded with a traveling load using the Green function method. During this process, the Volterra equations with delay were derived and solved, and the experimental results were in good agreement with the numerical analysis [4]. Moverover, Wickert [5] studied the nonlinear vibration and bifurcation of an axially moving beam and formed a complete framework of that. Vestroni [6] carried out the nonlinear dynamic behavior of a simply supported axially moving beam by the Galerkin method. On the basis of the previous researches, Pellicano and Vestroni [7] investigated the complex nonlinear dynamic behavior of a high-speed axially moving beam by means of a high-dimensional discrete model. Chakraborty and Mallik [8] addressed the free and forced vibration of a nonlinear axially moving beam which has an intermediate guide. Ghayesh and Balar [9] analyzed the parametric vibration and stability of an axially moving viscoelastic Rayleigh beam considering the geometric nonlinearity. Later, Ghayesh et al. [10] did the similar works in an axially moving laminated composite beam with the constant axial velocity. Huang [11] focused on the nonlinear vibration and bifurcation of an axially moving beam, which is subjected to a periodic lateral force excitation. The previous researches were mainly based on uniform axial moving, but there are also many ¨ et al. studies dealt with the axially moving beam with the time-dependent axial velocity. Oz [12] investigated the nonlinear vibration of an axially moving beam with the method of multiple scales, where the beam is moving with a time-dependent velocity. The stability of the structure ¨ under three different velocity functions are discussed. Ozhan [13] aimed to analyze the stability of an axially moving damped beam under the combined effects of variable speed and axial force. For the axially accelerating viscoelastic beam, Chen and Yang did the contribution. Chen et al. [14] analyzed the dynamic stability of a viscoelastic beam subjected to an axially accelerated speed. Later, considering the geometric nonlinearity, Yang et al. [15] re-derived the bifurcation and chaos of the accelerating viscoelastic beam. Moreover, for the effect of blast load on the vibration behaviors of the beam without the axial velocity, Zhai et al. [16] performed the characteristics of a reinforced concrete beam subjected to a blast load using the numerical and experimental method, respectively. Besides, there are a lot of researchers interested in the effect of blast load on reinforced concrete beam [17–20]. Nassr et al. [21] studied strength and stability of a steel beam column under blast load. Chernin et al. [22] presented an analytical method to predict the blast dynamic responses of a beam-column subjected to the combined effects of axial and transverse loads. Langdon [23] analyzed the dynamic responses of a composite sandwich structure with air-blast load using the ˇ experimental and numerical means. Sliseris et al.[24] established a nonlinear model to calculate 2
the dynamic behavior of cross laminated timber under the blast load. The numerical model was compared with classical beam theory and the results show that the shear deformation must be taken into account when designing CLT. Bodaghi et al. [25] studied the nonlinear active control of a functionally graded beam subjected to a blast load in thermal environment. Obviously, much research has been done about the effect of axial velocity on dynamics of an axially moving structure, and the influence of blast load on the beam without axial moving has been also studied by many researchers. However, the analysis for considering both effects of blast load and temperature or the combined effects of axial velocity, blast load, and temperature, are still far from perfect. To address the lack of that, we establish a mathematical model to investigate the nonlinear dynamic behavior of a symmetrically laminated beam subjected to the combined effects of blast load, axially moving and thermal environment. The rest of this paper is organized as follows. In Section 2, the second-order ordinary differential governing equation of a symmetrically laminated beam is established, and the beam with axial velocity is subjected to a blast load in thermal environment. In Section 3, the frequency of the nonlinear free vibration is obtained by using the multiple scale method. In Section 4, the relationship of the temperature variation and the axial velocity in critical condition is derived. In Section 5, the effects of different kinds of blast loads, axial velocity and temperature variation are discussed regarding to the nonlinear responses of the system. 2. Governing equations 2.1. Model of laminated beam Consider an axially moving laminated beam subjected to a blast load in thermal environment. The model of the beam is depicted in Fig. 1 and the cross section is shown in Fig. 2, in which l is the length, b is the width and H is the total thickness of the beam, vx is the axial velocity of the beam in x-direction, w(x, t) is the lateral deflection and P (t) is the blast load. The total number of the layers is n and the fibre orientation lies symmetrically with respect to the mid-plane of the beam. The thickness of the k − th layer is tk = zk − zk−1 .
Fig. 1. Model of the laminated beam
2.2. Basic equations
3
Fig. 2. Cross section model of the laminated beam
On the basis of the Euler-Bernoulli beam theory, the displacement field of the beam can be expressed as u1 = u − zw,x u2 = 0 u3 = w
(1)
where ui (i = 1, 2, 3) represent the displacements in x-, y- and z-direction at any point. u and w are axial and transverse displacements of the mid-plane. Employing the large-displacement assumption, the the axial strain of the beam can be obtained as εxx = ε0xx − zw,xx
(2)
where ε0xx represents the mid-plane strain and given as [26] 1 2 ε0xx = u,x + w,x 2
(3)
Considering the thermal effect of the composite material, the axial stress-strain relation of the present laminated beam model can be expressed as σxx = Q11 [εxx − αx ∆T ]
(4)
where αx = α1 cos2 θ + α2 sin2 θ
(5)
4
and Q11 = Q11 cos4 θ + 2(Q12 + 2Q66 ) sin2 θ cos2 θ + Q22 sin4 θ
(6)
where α1 and α2 are the thermal expansion coefficients of a single layer material along the fibre orientation and perpendicular to the fiber direction, respectively. ∆T is the variation of the temperature. θ is the fiber orientation angle and it is defined as the angle from x-axis to the fiber normal direction. Qij (i, j = 1, 2, 6) are elements of the stiffness of single layer material and have the following forms E1 E2 , Q22 = Q11 = (1 − ν12 ν21 ) (1 − ν12 ν21 ) E1 ν12 Q12 = , Q66 = G12 (1 − ν12 ν21 ) where E1 and E2 are elastic moduli of the material in principle directions and G12 is shear modulus, ν12 and ν21 are Poisson’s ratios.
2.3. Derivation of transverse governing equation The governing equation of motion can be obtained using the Hamilton’s principle, which can be expressed as ∫ t2 (δU − δV − δK)dt = 0 (7) t1
where δ is the variation operator, K is the kinetic energy, U is the potential energy, V is the work done by external force P (t). The variation of kinetic energy is written as δK =
∫ l∫ 0
ρ(uδ ˙ u˙ + wδ ˙ w)dAdx ˙
(8)
A
and the the variation of strain energy and the variation of work are defined as ∫ l∫ δU = σij δϵij dAdx 0
δV =
∫
(9)
A
l
P (t)δwdx
(10)
0
where ρ is the material density and A is the cross-sectional area of the beam. Then, we define the axial resultant Nxx and the bending moment Mxx in unit width as follows
5
[ ] Nxx , Mxx =
∫
A
[ ] σxx 1, z dA
(11)
And then Eqs. (8) and (9) can be rewritten as δK =
δU =
∫ ∫
l
ρA(uδ ˙ u˙ + wδ ˙ w)dx ˙
(12)
[Nxx (δu,x + w,x δw,x ) − Mxx δw,xx ]dx
(13)
0
0
l
Substituting Eqs. (10), (12), and (13) into Eq. (7), the governing equations with respect to axial resultant and bending moment can be obtained Nxx,x = ρA¨ u
(14)
Nxx,x w,x + Nxx w,xx + Mxx,xx + P (t) = ρAw¨
(15)
On the basis of Ref. [6], the term ρAw¨ of the axially moving beam can be given as ρAw¨ = ρA(
2 ∂ 2w ∂ 2w 2∂ w + 2v + v ) x x ∂t2 ∂x∂t ∂x2
(16)
Combining Eqs. (2), (4) and (11), one can derive 1 2 Nxx = A11 (u,x + w,x ) − B11 w,xx − NxT 2 1 2 Mxx = B11 (u,x + w,x ) − D11 w,xx − MxT 2
(17)
where A11 , B11 and D11 are the tensile, stretching-bending coupling and bending stiffness, respectively. They are denoted as [ ] A11 , B11 , D11 =
∫
H/2
−H/2
[ ] Q11 1, z, z 2 dz
(18)
NxT and MxT are the additional membranous force and the additional bending moment generated
6
by the temperature variation. They are defined as [ T ] Nx , MxT =
∫
H/2
−H/2
[ ] Q11 1, z αx ∆T dz
(19)
Assuming the in-plane axial inertia force is to zero, the axial resultant in Eq. (14) can be expressed as a function independent of x. Then integrating Eq. (17) into Eq. (14) and solving u yields 1 u=− 2
∫
x
0
w,ξ2 dξ +
1 T 1 B11 w,x + Nx + b1 (t)x + b2 (t) A11 A11 A11
(20)
where b1 (t) and b2 (t) are the integral undetermined term. The axial displacement boundary conditions are shown as u|x=0 = 0 u|x=L = 0
(21)
Substituting Eq.(20) into Eq. (21), the b1 (t) and b2 (t) can be obtained ∫ A11 l 2 B11 b1 (t) = w,x dx − [w,x (l) − w,x (0)] − NxT 2l 0 l B11 b2 (t) = − w,x (0) A11
(22)
with the help of Eqs. (20) and (22), the Eq. (15) can be rewritten as ∫ 2 B11 A11 l 2 T (D11 − )w,xxxx + [Nx − w dx]w,xx A11 2l 0 ,x ∂2w ∂ 2w ∂ 2w = P (t) − ρA( 2 + 2vx + vx2 2 ) ∂t ∂x∂t ∂x
(23)
2.4. Galerkin discretization The governing equation given by Eq. (23) can be reduced in time domain by choosing the approximation function for displacement field and employing the Galerkin method. The approximate solution is usually represent as [6]. w(x, t) =
M ∑
Um (t)ϕm (x)
(24)
m=1
where ϕm (x) is the trial function, M is the truncation order, and the choice of the trial function depends on the boundary condition. The trial functions for different boundary conditions are 7
shown as Hinged-Hinged(HH): w|x=0,l = 0; w,xx |x=0,l = 0 ϕm (x) = sin(
(25)
mπx ) l
Clamped-Clamped(CC): w|x=0,l = 0; w,x |x=0,l = 0 ϕm (x) = cos(km x) − cosh(km x) + [
sin(km l) + sinh(km l) ][sin(km x) − sinh(km x)] cos(km l) − cosh(km l)
(26)
Frequency equation(CC): cos(km l) · cosh(km l) = 1 Clamped-Free(CF): w|x=0 = 0; w,x |x=0 = 0 ϕm (x) = cos(km x) − cosh(km x) − [
cos(km l) + cosh(km l) ][sin(km x) − sinh(km x)] sin(km l) − sinh(km l)
(27)
Frequency equation(CF): cos(km l) · cosh(km l) = −1 For the general result, introducing Eq. (24) into Eq. (23) and employing the Galerkin principle, a set of ordinary differential equations can be obtained M ∑
mmi U¨i +
i=1
M ∑ i=1
cmi U˙ i +
M ∑ i=1
kmi Ui −
M ∑ M ∑ M ∑
nl kmikl Ui Uk Ul = pm
(28)
i=1 k=1 l=1
where mmi = ρHb
∫
l
ϕi (x)ϕm (x)dx ∫ l 2 ∫ l B11 ′′′′ T 2 kmi = (D11 − ) ϕ (x)ϕm (x)dx + (Nx + ρHbvx ) ϕ′′i (x)ϕm (x)dx A11 0 i 0 ∫ l ϕ′i (x)ϕm (x)dx cmi = 2ρHbvx 0 ∫ l pm = P (t)ϕm (x)dx 0 ∫ ∫ l A11 l ′′ nl ϕm (x)ϕi [ ϕ′k (x)ϕ′l (x)dx]dx kmikl = 2l 0 0 0
8
(29)
3. Nonlinear free vibration The multiple scales method is adopted to solve the nonlinear free vibration equation that is Eq. (28) without the external force, and it can be rewritten as follows U¨j + Cj U˙ j + ωj2 Uj −
M ∑ M ∑ M ∑
nl gjmkl Um Uk Ul = 0
(30)
m=1 k=1 l=1
where ωj is the linear frequency that can be obtained from the mass matrix and stiffness matrix. In the light of nonlinear dynamics theories in Ref. [27], the expression of Uj , its derivatives ˙ Uj and U¨j are given as Uj =εUj1 (T0 , T1 , T2 ) + ε2 Uj2 (T0 , T1 , T2 ) + ε3 Uj3 (T0 , T1 , T2 ) + ... U˙j =ε(D0 Uj1 ) + ε2 (D0 Uj2 + D1 Uj1 ) + ε3 (D0 Uj3 + D1 Uj2 + D2 Uj1 ) + ... U¨j =εD2 Uj1 + ε2 (D2 Uj2 + 2D0 D1 Uj1 )+ 0
(31)
0
ε3 Uj3 (D02 + 2D0 D1 Uj2 + D12 Uj1 + 2D0 D2 Uj1 ) + ... where Di = ∂/∂Ti , the small parameter ε is a book-keeping device, T0 , T1 and T2 are different time scales which can be expressed as t, εt and ε2 t, respectively. Substituting Eq. (31) into Eq. (30) collecting terms of the same order of ε yields Order ε1 : D02 Uj1 + ωj2 Uj1 = 0 f or
j = 1, 2, ...
(32)
Order ε2 : D02 Uj2 + ωj2 Uj2 = −2D0 D1 Uj1
f or
j = 1, 2, ...
(33)
Order ε3 : D02 Uj3 + ωj2 Uj3 = −
D12 Uj1
− 2D0 D2 Uj1 − 2D0 D1 Uj1 +
M ∑ M ∑ M ∑
nl gjmkl Um Uk Ul
f or
j = 1, 2, ...
(34)
m=1 k=1 l=1
Assuming the general solution of Eq. (32) is [27] Uj1 = Aj (T1 , T2 )eiωj T0 + CC
(35)
where Aj (T1 , T2 ) is a function to be determined, CC is the complex conjugate of the preceding term. Substituting Eq. (35) into Eq. (33) yields D02 Uj2 + ωj2 Uj2 = −2iωj D1 Aj (T1 , T2 )eiωj T0 + CC 9
(36)
Eliminating the secular term of Eq. (36), we have −2iωj D1 Aj (T1 , T2 ) = 0
(37)
Thus it can be seen, Aj (T1 , T2 ) is only a function of T2 . Inserting Eq. (37) into Eq. (36) derives Uj2 = 0
(38)
Substituting Eq. (35), Eq. (36) and Eq. (37) into Eq. (34), the following equation can be given by D02 Uj3 + ωj2 Uj3 = − 2iωj D1 A′j (T2 )eiωj T0 +
nl gjjjj [A3j (T2 )e3iωj T0 + 3A2j (T2 )Aj (T2 )eiωj T0 ] + cc
(39)
Similarly, eliminating the secular term of Eq. (39), we have nl −2iωj A′j (T2 ) + 3gjjjj A2j (T2 )Aj (T2 ) = 0
(40)
Letting Aj = (1/2)αj (T2 )eiβj (T2 ) , and inserting it into Eq. (40), two differential equations can be obtained as d αj (T2 ) = 0 dT2
(41)
d 3 nl 2 βj (T2 ) = − gjjjj a dT2 8
(42)
The solutions of Eqs. (41)-(42) are as follow αj (T2 ) = αj0 ,
(43)
3 nl 2 βj (T2 ) = − gjjjj a T2 + βj0 8
(44)
where aj0 and βj0 are constants, which depend on the initial conditions. Substituting Eqs. (43)(44) into the expression of Aj , and inserting the result into Eq. (35), the nonlinear frequency can be denoted ωnj = ωj −
3 nl 2 g a . 8ωj jjjj j0
(45)
10
4. The stability region In order to provide the theoretical basis for subsequent analysis, the relationship of the axial velocity and the temperature variation in critical condition must be determined first. For a simply supported symmetrically laminated beam, the trial function ϕm (x) is given by Eq.(25). Taking the truncation term M = 1 and substituting Eq. (25) into Eq. (28) of which the nonlinear term and the external force is not included, thus the system equation can be expressed as M1 U¨1 + C1 U˙1 + K1 U1 = 0
(46)
where ∫
l ρAl , M1 = ρHb ϕ21 (x)dx = 2 0 ∫ l ∫ l (4) (2) T 2 K1 = D11 ϕ1 (x)ϕ1 (x)dx + (Nx + ρHbvx ) ϕ1 (x)ϕ1 (x)dx 0 4
0
2
π D11 π − (NxT + ρAvx2 ) , = 3 2l 2l ∫ l C1 = 2ρHbvx ϕ′1 (x)ϕ1 (x)dx = 0.
(47)
0
Then we arrive at ω12
D11 π 4 π2 T 2 = K1 /M1 = − (Nx + ρAvx ) ρAl4 ρAl2
(48)
It satisfies the following inequality D11 π 4 π2 T 2 − (N + ρAv ) ≥0 x x ρAl4 ρAl2
(49)
Hence, we can derive C∆T +
ρAvx2
D11 π 2 ≤ ρl2
(50)
where C=
n ∑ k=1
(Q11 αx + Q12 αy + Q16 αxy )k (zk − zk−1 )
11
(51)
The stability region R is defined as } { 2 D π 11 R = (∆T, vx ) |C∆T + ρAvx2 ≤ ρl2
(52)
For arbitrary values of the axial velocity and the temperature variation belonged to the region D11 π 2 2 is the critical (∆T, vx ) ∈ R, the motion is stable. Physically speaking, C∆T + ρAvx = ρl2 condition for instability. Similarly, taking the truncation term M = 2 and following the step described above, we obtained ¨ + C∗ U ˙ + K∗ U = 0 M∗ U
(53)
where
ρAl 0 [m 0 ] 2 ∗ M = = ρAl 0 m 0 2
(54)
(55)
D11 π 4 π2 [ ] T 2 0 k11 0 2l3 − (Nx + ρAvx ) 2l ∗ K = = 8D11 π 4 0 k22 π2 T 2 0 − 2(N + ρAv ) x x l3 l
C∗ =
0 8 ρAvx 3
8 − ρAvx [0 −c] 3 = c 0 0
(56)
The characteristic equation of the system is [27] 2 ∗ λ M + λC∗ + K∗ = 0
Solving the Eq. (57), we can readily derive √ √ −B− A λ1,2 = ± i 2m2 √ √ −B+ A i λ3,4 = ± 2m2
(57)
(58)
12
where 2
A = B − 4k11 k22 m2
B = k11 m + k22 m + c2 According to Eq. (58), we can arrive at {
} 2 D π 11 R1 = (∆T, vx ) |C∆T + ρAvx2 ≤ l2 { } 2 4D π 11 R2 = (∆T, vx ) |C∆T + ρAvx2 ≤ l2
(59)
where R1 and R2 are the stability regions of the fundamental frequency and the high frequency, respectively. As we can see, the region R1 is a subset of the region R2 . On the analogy of this, we can find that the low frequency stability region is a subset of the hight frequency stability region as the value of the truncation term M is greater. For the other various boundary conditions the corresponding stable domain can be obtained by similar methods. 5. Numerical results In this section, the numerical results for the symmetrically laminated beam subjected to a blast load in thermal environment are performed by using the fourth-order Runge-Kutta method. However, considering the influence of the axial velocity, the one-term approximation formula for the displacement function is not applicable to this study. Therefore, how to determine the truncation order M is also a mission in this paper. The elastic modulus of graphite/epoxy with various temperature is given in Table 1 [28] and the geometric parameter in Table 2 are taken as an example in the following analysis. The density of the beam is ρ = 1600kg/m3 , the ply angle is (0◦ /90◦ )s . The thermal expansion coefficients and Poisson’s ratio are taken as α1 = −0.3 × 10−6 , α2 = 28.1 × 10−6 , and v21 = 0.3. Table 1 Elastic moduli of graphite/epoxy composite.
Moduli(GPa)
Temperature, T (◦ C) 25
50
75
100
125
150
E1
130
130
130
130
130
130
E2
9.5
9.0
8.5
8.0
7.5
6.75
G12
6.0
6.0
5.5
5.0
4.75
4.5
In Section 2, ∆T = T − T0 is defined as temperature variation. To unify the discussion, we define T0 = 25◦ C as referenced values in the present study. 13
Table 2 Geometric parameters of the laminate beam
Length(m)
Width(m)
Thickness(m)
Layers
1
0.05
0.0125
4
5.1. The critical curve Based on Eq. (50) and Eq. (59), the one-two orders critical curves about the axial velocity vx and the temperature variation ∆T can be obtained, which are shown in Fig. 3 and Fig. 4.
Fig. 3. One orders critical curve
Fig. 4. Two orders critical curve
As indicated in Fig. 3 and Fig. 4, the lines illustrate the relationship between the temperature variation ∆T and the axial velocity vx in the critical state. In other words, the area below the line is the stability region as defined in Section 4. Noticeably, the vibration of the laminated beam is stable as the values of ∆T and vx are in this area. Comparing Fig. 3 and Fig. 4 shows that the stability region of the high-order frequency is larger than that of the low-order. Generally speaking, in order to meet the requirement, the values of ∆T and vx should be in the stability region of the low-order. This phenomenon is similar to the result in Section 4. 5.2. Convergence validation To determine the truncation order M , the frequency of the laminated beam under different boundary conditions with different M are performed. The blast load is an uniform pressure, which is selected the explosive blast load as an example in this subsection, and it is given by [29] { Pm (1 − t/tp )e−ηt/tp P (t) = 0
0 ≤ t ≤ tp t > tp
(60)
14
where Pm is the peak blast pressure, tp is the positive phase duration, η is the waveform parameter. For the next calculation, the parameters in Table 3 are adopted. Table 3 Relevant parameters for calculation
Pm (MPa) 3.447
tp (s) 0.02
η 2
vx (m/s) 50
∆T (◦ C) 30
The influence of the truncation order M on the frequency is discussed under different boundary conditions by using the eigenvalue method and it is shown in Fig. 5. The HH, CC, and CF represent the boundary conditions as mentioned in section 2.4.
(a)
(b)
(c)
Fig. 5. Variation of frequency with truncation order M under different boundary conditions
It can be found from Fig. 5 that the frequencies of the three boundary conditions (HH, CC, and CF) achieve convergence as the M takes 10. Hence the truncation order M can be considered as 10 for the following analysis. 5.3. Comparison validation In order to investigate the accuracy of the above prediction, a finite element simulation model is carried out employing LS-DYNA software. The hexahedral elements with 20 nodes are used in the modeling process and the total number of elements is 1000. The response at the centre of the simply supported laminated beam is solved by using the Runge-Kutta method developed in this paper, in which the parameters of axial velocity, positive phase duration, maximum blast pressure, and temperature variation are the same with the previous case. The result is compared with that obtained by the finite element method (FEM), and the comparison is presented in Fig. 6. It indicates that the two sets of results are in good agreement with each other. 5.4. Results and discussions Firstly, for the simply supported symmetrically laminated beam, the effect of the axial velocity on the variation of the response is detailed in Fig. 7. It can be observed that the variation of the response is imperceptible in time domain [0,0.01]s and obvious in [0.01,0.02]s. In order 15
Fig. 6. Responses of the present and FEM
to facilitate observation, Fig. 8 gives the variation of the response versus time in time domain [0.01,0.02]s particularly.
Fig. 7. Effect of the axial velocity on the flexural amplitude
It is evident from the result that the flexural amplitude decreases while the axial velocity increases due to the gyro-effect. The reasons are as follows: from Eq. (23), the axial velocity affects the stiffness and damping of the structure, but we don’t know which one has a greater impact on the response. Therefore, the following discussions are made. Letting Kk and Cc instead of the terms ρAvx2 and ρAvx in Eq. (23) yields 3 2 − NxT − Kk )w,xx = −P (t) −2Cc w,xt − ρAw,tt − D11 w,xxxx + ( A11 w,x 2
(61)
Based on this equation, four cases are considered, including: (1) Kk = 0, Cc = 0, this case is not taken into account the axial velocity; (2) Cc = 0, Kk ̸= 0, which means the axial velocity only affects the stiffness; (3) Kk = 0, Cc ̸= 0, it represents the axial velocity only affects the damping; (4) Kk ̸= 0, Cc ̸= 0, it is the model studied in this paper. The amplitude responses of the four cases are shown in Fig. 9. As you can see in Fig. 9, the amplitude response of the second case is more nearly the model studied in this paper, which means the influence of 16
Fig. 8. Variation of flexural amplitude versus time in the range of 0.01s to 0.02s
damping on response is greater than that of stiffness. Therefore, the flexural amplitude decreases while the axial velocity increases due to the gyro-effect, and the greater the velocity, the greater the damping. In addition to, as the the axial velocity increases, the period of vibration becomes longer.
Fig. 9. Variation of flexural amplitude for the four cases versus time
On the other hand, the motion state of the system changes into nonlinear free vibration while t > ηtp as mentioned by Eq. (60), and it is listed in Fig. 10. The datum in Fig. 10 suggest that the nonlinear response and frequency of free vibration are related to the initial condition and linear frequency, which is agreement with the Eq. (45). Secondly, the time history of the amplitude at the centre of the laminated beam with different temperatures is shown in Fig. 11. It explains that the vibration period and amplitude of the laminated beam increases with the increase of the temperature. As we know, the temperature has an influence on the stiffness of the structure, and the higher the temperature variation, the smaller the stiffness will be. In the main, the smaller the stiffness, the lager the flexural amplitude. So we can found from Fig. 11 that with increasing the temperature, the flexural 17
Fig. 10. Effect of the axial velocity on the nonlinear free vibration
amplitude increases, and the frequency decreases.
Fig. 11. Effect of the temperature on the flexural amplitude
Fig. 12 gives the response at the centre of the laminated beam versus time with different positive phase durations. As evident from Fig. 12, the increase of the positive phase duration tp leads to the increase of the amplitude. Physically speaking, as the tp increasing, the blast load subjected to the beam will be lasted for a longer time so that the amplitude will be increased. Furthermore, the period of vibration is found to be shorten with increasing positive phase duration. Fig. 13 illustrates the influence of different peak blast pressures on the response at the centre of the laminated beam. As can be seen from the approximate-numerical result in Fig. 13, the flexural amplitude and frequency increase while the peak blast pressure increases. Then, a comparison of the deflection time history of the laminated beam with different ply angles is shown in Fig. 14. It indicates that as the ply angle is (0◦ /0◦ )s , the flexural amplitude is the minimum. On the contrary, when the ply angle is (90◦ / − 90◦ )s , the flexural amplitude reaches the maximum. In addition, the vibration response of the laminated beam with different boundary conditions under explosive blast load is shown in Fig. 15. As we can see from the Fig. 15, when the boundary condition is CF, the amplitude and 18
Fig. 12. Effect of the positive phase duration on the flexural amplitude
Fig. 13. Effect of the peak blast pressure on the flexural amplitude
Fig. 14. Effect of the ply angle on the flexural amplitude
19
Fig. 15. Effect of the boundary condition on the flexural amplitude
the period of the vibration are the biggest, while the amplitude and the period are the smallest when the boundary condition is CC. Furthermore, there are various types of blast loads. In the following analysis, the effects of different blast loads on the responses of the laminated beam are discussed. Firstly, the triangular load is given by [29] { Pm (1 − t/tp ) 0 ≤ t ≤ rtp P (t) = 0 t < 0 and t > rtp
(62)
where r indicates the shock pulse length factor. The shape of the pulse is decided by r. For the case of r = 1, it is called air-blast, for r = 2, it is antisymmetric about x-axis and called sonic boom. Substituting Eq. (62) into Eq. (28) and solving it, the results for the case of r = 1 with different axial velocities and temperatures are shown in Fig. 16 and Fig. 17, respectively.
Fig. 16. Effect of the axial velocity on the response of the beam subjected to the air-blast
As can be observed in Fig. 16 and Fig. 17, there is a slight changes of the flexural amplitude with the increasing axial velocity and temperature, which means that a single factor, the axial velocity or the temperature has relatively limited influence on the flexural amplitude and the 20
Fig. 17. Effect of the temperature on the response of the beam subjected to the air-blast
vibration frequency. What’s more, it also shows that the air-blast has a greater impact on the structure than the explosive blast load. Secondly, the numerical result for laminated beam subjected to the nuclear blast load is performed. The function of the pulse is given by [29] { Pm (1 − t/tp ) 0 ≤ t ≤ t1 P (t) = P0 (1 − t/t2 ) t1 ≤ t ≤ t2
(63)
This pulse is consist of two linear functions, the Pm is the same with the above and we choose the P0 = 0.5Pm , tp = 0.01s, and t2 = 0.02s for the function as an example, and the t1 is (Pm − P0 )(tp t2 )/(Pm t2 − P0 tp ). For the reason that the pulse is in accordance with Eq. (62) as the time in the range of 0 < t < t1 , hence we mainly focus on the response for the second part of the pulse and its results are presented in Fig. 18 and Fig. 19. It can be identified that the amplitude decreases/increase while the axial velocity/temperature increase.
Fig. 18. Effect of the axial velocity on the response of the beam subjected to the nuclear pulse
In the end, the dynamic response of the laminated beam subjected to the sine pulse is 21
Fig. 19. Effect of the temperature on the response of the beam subjected to the nuclear pulse
analyzed, and the sine pulse can be expressed as [29] { Pm sin(πt/rtp ) 0 ≤ t ≤ rtp P (t) = 0 t < 0 and t > rtp
(64)
The pulse is a periodic excitation, therefore, in this case, the amplitude and the vibration frequency of the beam mainly depend on the rtp of the periodic excitation. Based on this, the axial velocity and the temperature hardly have any influence on the flexural amplitude and the vibration frequency, as seen in Fig. 20 and Fig. 21.
Fig. 20. Effect of the axial velocity on the response of the beam subjected to the sine pulse
6. conclusion In this work, the nonlinear dynamic response of a symmetrically laminated beam is studied under the framework of geometric nonlinearity, and the combined effects of the axial velocity, blast load and thermal environment are considered. The nonlinear coupled equation in the time domain is obtained by employing the Galerkin method. The truncation order of the approximate 22
Fig. 21. Effect of the temperature on the response of the beam subjected to the sine pulse
solution is selected by convergence calculation of the dynamic response solved by the RungeKutta method. Through numerical analyses, the effects of some detailed parameters of the laminated beam on the nonlinear dynamic response are discussed. The results of this work can be summarized as follow: 1) For nonperiodic blast load: a) Due to the gyro-effect, the flexural amplitude and frequency decrease while the axial velocity increases. b) With increasing temperature, the flexural amplitude increases, and the frequency decreases. 2) For the explosive blast load: a) As the positive phase duration tp increases, the flexural amplitude and frequency tend to increase. b) The flexural amplitude and frequency increase while the increase of the peak reflected pressure. c) As the ply angle is (0◦ /0◦ )s , the flexural amplitude is the minimum, when the ply angle is (90◦ / − 90◦ )s , the flexural amplitude reaches the maximum. 3) For the sine pulse, the amplitude and vibration frequency of the beam mainly depend on the periodic excitation. Namely, the axial velocity and the temperature hardly have any influence on the flexural amplitude and vibration frequency. 4)When the boundary condition is CF, the amplitude and the period of the vibration are the biggest, while the amplitude and the period are the smallest when the boundary condition is CC. Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 11372257, 11472064, 11602208, 51674216), the Science and Technology Foundation of ChongQing Education Commission(Nos. KJ1500929), the project of Chongqing University of Science and Technology (Nos. Shljzyh2017-007). The supports from the innovative cultivation fund of the excellent doctoral graduate students of Southwest Jiao Tong University are gratified as well. References [1] H Ashley and G Haviland. Bending vibrations of a pipe line containing flowing fluid. Journal of Applied Mechanics-Transactions of the ASME, 17(3):229–232, 1950.
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Highlights
The geometrical nonlinearity of the structure is considered.
The axial velocity, temperature and blast load are considered at the same time.
Effects of axial velocity and temperature on the nonlinear response are discussed.