Dynamic Behavior of a Functionally Graded Beam Under a Moving Load on Nonlinear Viscoelastic Foundation Considering Moving Mass

Nội dung text: Dynamic Behavior of a Functionally Graded Beam Under a Moving Load on Nonlinear Viscoelastic Foundation Considering Moving Mass

1. Tuyển tập Hội nghị khoa học toàn quốc lần thứ nhất về Động lực học và Điều khiển Đà Nẵng, ngày 19-20/7/2019, tr. 261-268, DOI 10.15625/vap.2019000288 Dynamic Behavior of a Functionally Graded Beam Under a Moving Load on Nonlinear Viscoelastic Foundation Considering Moving Mass Huynh Van Quang Faculty of Architecture and Civil Engineering, Binh Duong University, Binh Duong, Vietnam E-mail: hvquang@bdu.edu.vn Abstract Based on the above discussions, there have been In this paper, linear dynamic analysis of a functionally graded several investigated on free vibration analysis of materials (FGMs) beams on nonlinear viscoelastic foundation functionally graded beams. Free vibration analysis of FG under to moving load considering moving mass has been beams based on Timsheko theory and Hamiltonian investigated. The nonlinear behavior of viscoelastic foundation principle, examined the effects of different boundary conditions, material distribution coefficient and is represented by the third order relationship. The governing oscillation mode to conduct FGMs beams [1]. Nonlinear equation of motion of the beams is derived based on Hamilton dynamic analysis of FGMs beams subjected to moving principle expressed as Lagrange’s equations with specific harmonic loads based on Timoshenko theory combining boundary conditions satisfied with Lagrange’s multiplier and deformation and nonlinear relationship Von-Kerman and sporadic by high order polynomial. The computer program equations of motion of the system obtained from the using Newmark- time integration and Newton Raphson equation Lagranges, vertical and horizontal procedure is written by MATLAB language. Material properties displacements are approximated by polynomial forms [2]. Vibration analysis of FGMs beams in different theory of the beam vary continuously in thickness direction according subjected to moving mass and the effect of shear to a power law form. The effects of the material distribution, deformation, the inertia, Coriolis, the moving mass on the velocity of the moving load, mass of load, parameters of dynamic displacements and the tresses of the beam [3]. foundation as linear, shear and nonlinear layer on the Research Euler beam under to moving harmonic load by displacement of the beam have been examined in detail. the finite element method, influence of the coefficient Keywords: Beams, Functionally graded beams, Nonlinear foundation and frequency of excitation force on the foundation, Moving mass, Dynamic analysis. beams are investigated [4]. The dynamic behavior of simply-supported isotropic beams subjected to an 1. Introduction eccentric axial load and a moving harmonic load was analyzed by using Euler–Bernoulli, Timoshenko and the Today, the development of computer science and the third order shear deformation theory [5], [6]. increasing demand of people requires materials with high Fundamental frequency analysis of FG beams by durability, application. Therefore, scientists who Japan different beam theories [7]. The investigation of the proposed multiple layers of composite material that is effects of inertial, centripetal, and Coriolis forces on the functionally graded material. energy in 1980s. In terms of dynamic response of a cracked beam under moving mass mechanics, multi-layer composite material has the load [8]. Dynamic response of a simply supported beam advantages over base materials to create it because they subjected to moving masses is considered [9]. are distributed in the appropriate position with the More recently, there have been few investigated on features of each type of material. FGMs are special effects of different parameters of elastic foundation on composites that have a continous variation of material behavior of FG beams [10], [11]. Investigation of the properties from one surface to another. These new kinds dynamic stability of (FGSW) and (FGO) by FEM on of material have been employed for a wide range of elastic foundation [12]. Dynamic responses of beam applications such as defense industries, aircrafts, space subjected to moving load and moving mass supported by vehicles, etc. Studies devoted to understand the static and Pasternak foundation based on Euler-Bernoulli [13],[14]. dynamic behavior of the FGM beams on the ground Along with the development of computer science, under dynamic loads is one of the problems attracting humankind has proposed many models with different many researchers around the world. This problem is foundation are more parameters recommended for highly practical, to describe behavior for many structures accurate results, close to reality [15]. The multi-parameter such as roads, railways, airport landing surfaces. To foundation model was created to describe nonlinear survey behavior for many on the above structure, we need behavior in the process of working with the above to select the model of beams and ground soil to suit the structure. structural analysis problem.
2. Huynh Van Quang b : width of the cross section : transverse displacement of any point on the h : depth of the cross section neutral axis : the area of the cross-section :axial displacement of any point on the neutral : the eigenvalues of the beam axis : dimensionless natural frequency : generalized coordinates : lagrange multipliers : matrices due to Lagrange multipliers : z coordinate : unit step function : x coordinate : time-dependent generalized coordinate of the axial : number of the terms in the displacement displacements functions :time-dependent generalized coordinate of the : mass matrix of the beam transverse displacements : length of the beam :time-dependent generalized coordinate of the : kinetic energy of the beam rotation displacements : moment of inertia of the cross section :normal stress : transverse displacement :normal strain : velocity of the moving mass :location of the concentrated moving harmonic load : potential of the external loads :linear stiffness matrix : strain energy of the beam : nonlinear stiffness matrix : axial displacement : inertial coefficients : power-law exponent : generalized load vector : lagrangian functional of the problem : extensional rigidity : material properties of the top surfaces : material properties of the bottom surfaces : the volume fractions of the top surfaces : the volume fractions of the bottom surfaces : the rotation of the cross-sections (z) :the characteristic shape function of horizontal : the shear strain slip and stress distribution along the beam height : dimension linear layer parameters : stiffness of shear foundation : dimension of shear layer parameter : dimension of nonlinear layer parameter : drag coefficient tipping : ratio of resistance BC : boundary conditions : acceleration due to gravity : the location of the moving mass :the velocity of the moving mass g :the gravitational acceleration c(t) :the unit step function t1 :the time when the mass M just comes t2 :the time when the moving mass M just leaves the onto the beam beam and also suggested in recent times. Dynamic analysis of Recently, a number of third-order nonlinear foundation beam on a new foundation model nonlinear behavior models proposed the relationship between force and under to a moving mass [23]. However, the behavior of displacement in the structural dynamics problem on the the beams on the nonlinear viscoelastic foundation under ground, giving results close to reality. a moving load considering moving mass has not been Confirmed through comparative research results turn studied much, especially the application materials in the deformed rails foundation model of linear, nonlinear and form of structural FGMs has not mentioned much. The experimental foundation fact [16]. In addition to aim of this paper is to investigate behavior of FGMs analyzing nonlinear behavior foundation, it makes more beams on nonlinear viscoelastic foundation under a viscous review, to describe more realistically. Most moving load. Newmark and Newton Raphson method is recently, analyzed the dynamic behavior of beams on the applied to the equations of motion of the beams. Also, nonlinear foundation under to load [17,18]. The stability examine the influence of parameters such as the effects of analysis of beam cross section changes based nonlinear the material distribution, velocity of the moving load, elastic foundation subjected to moving load by FEM, the parameters of foundation as linear, shear, nonlinear layer results indicate the influence of the boundary conditions and mass load on the dynamic responses of the beam are of the beam with parameters different foundation to the discussed in detail. vertical displacement of the beam [19]. Studying dynamic behavior of FGM beams based on nonlinear elastic 2. Functionally graded materials substrates including shearing layer and cubic nonlinearity We consider a FGMs: L, b, h, with coordinate system and effect of elastic foundation coefficients associated (Oxyz) having the origin O and respectively, resting on with boundary conditions and nonlinear behavior [20]. an nonlinear viscoelastic foundation under to a moving Study nonlinear behavior of FG beams on nonlinear load considering moving mass is show in Fig. 1. elastic foundation under to axial force based on Euler-bernoulli [21]. The nonlinear equations of motion are solved via Newmark- method [22]. Through this, the foundation model is being researchers worldwide interest
3. Dynamic behavior of a functionally graded beam under a moving load on nonlinear viscoelastic foundation Considering Moving Mass 2 2  u0 u0  w0 u0  0 Axx 2 Exx Bxx 2 2Exx x x x x x 2 1 2w 2w  U D H 2F 0 2 F H 0 0 dAdx. xx xx xx 2 xx xx 2  2 L x x x 2 w0 Axz 0 x  (8) Fig. 1. A FGMs beams on nonlinear foundation A , B , D E(z)(1, z, z 2 )dA due to moving mass M. xx xx xx A where (E , F ) (z)E(z)(1, z)dA (9) xx xx It is assumed that the FGMs beams is made of A 2 ceramic and metal. According to the rule of mixture, the (H ) (z) E(z)dA xx  effective material properties [2]: A 2 k (z) z (Axz ) G(z)dA (1) z P(z) (Pc Pm ) 0.5 Pm A h The deformation energy in the nonlinear foundation is Based on the third order shear deformation theory, the axial displacement, and the transverse displacement 2 1 1 w 1 of the beam, . U k w2dx k dx k w4dx F 2 L 2 G x 4 NL L L L (10) (2a) The kinetic energy of the beam at any instant is (2b) 1 K (z)(v 2 v 2 )dAdx (11) 2 x z Timoshenko beam theory (FSDBT): (z) z (3) V 2 u u  w  The strain-displacement relations are given by: I 0 2(I I ) 0 0 A t E B t t x 2  w0 (4a) (I D I H 2I F ) 1 t x K dx 2  2 L  w0  0  0 (4b) 2(I F I H ) I H t x t t 2 u  w 2I 0 0 I 0 By assuming that the material of FGMs beams obeys E A t t t  Hooke’s law, the stresses in the beam become: (12) By defining the following cross-sectional inertial (5a) coefficients: (I , I , I ) (z)(1, z, z2 )dA (13a) (5b) A B D A (I ,I ) (z)(1, z)dA (13b) By neglecting the rotary inertia of the moving mass, E F A the kinetic energy of the moving mass is (I ) [(z)]2 (z)dA (13c) H A (6) Potential of the moving load at any instant [3]: V Mg.w (x ,t)[c(t t ) c(t t )] (14) 0 m 1 2 The strain energy of the beam is The Lagrange multipliers formulation of the considered problem requires constructing the Lagrangian 1 U     dAdx (7) functional as follows: 2 xx xx xz xz L
4. Huynh Van Quang The equations of motion (20) written as compact: w G w ( L/2,t) w (L/2,t) 0 ( L/2,t) s 1 0 2 0 3 M MS q (t) C 2Mv H  q (t) x i m i L 2 NL (21) w0 (15) K Mv R K q(t) q(t) P (t) (L/2,t) u ( L/2,t) u (L/2,t) m  M 4 x 5 0 6 0 ( L/2,t) (L/2,t) T 7 0 8 0 where q(t) A(t), B(t),C(t), (t) are the The functional of the problem is: generalized coordinates. The following frequency equation can be expressed in the following matrix form:  K K G (U U ) V (16) M s F K  q  2 M q 0 (22) The unknown functions, are approximated by space dependent polynomial terms, The following non-dimensional frequency [15]. . ; (23) (17) 3. Simulation results Then using the Lagrange’s equations given: In the numerical results, dynamic responses of a (18) FGMs beams on nonlinear viscoelastic foundation due to (19a) moving loads has been investigated. (19b) (19c) Table 1: Material property of FGMs constituents. q ,q ,q ,q , Property Unit Aluminum Zirconia 3N 1 1 3N 2 2 3N 3 3 3N 4 4 (19d) q ,q ,q ,q E GPa 70 200 3N 5 5 3N 6 6 3N 7 7 3N 8 8 kg/m3 2700 5700 Yields the following coupled non-linear systems of  - 0.3 0.3 equations of motion: Table 2: The dimensions and attribute of the beam.  L (m) b (m) h (m) ks k M11 M S1 N N 0 N N 0 N N 0 N 8 An(t)  20 0.5 0.5 5/6 1 0 N N  M22  M S2 N N M23 N N 0 N 8 B (t) n  0 M M 0  N N 32 N N 33 N N N 8 Cn(t) Table 3: The characteristic of foundation 0 0 0 0  8 N 8 N 8 N 8 8 n(t) K1 K2 K3 (%)  10 5 20 5 C11 2Mvm H1 N N 0 N N 0 N N 0 N 8 An (t)  Table 4: Properties of load 0 N N 0 N N 0 N N 0 N 8 B (t) n  0 0 0 0  P(kN) mr v(m/s) N N N N N N N 8 Cn (t)  1500 1 20 0 8 N 0 8 N 0 8 N 0 8 8 n (t) The following non-dimensional variables: L W 2 L L S K11 K11 Mvm[R1] N N K12 N N K13 N N K14 N 8 An(t) 2 4 kG L k L L L L L S C 2 k A K K B (t) cr l m 2 2 1 K21 N N K22 N N K23 N N K24 N 8 n EI EI  L L L S 6 K K K K Cn(t)  k L 31 N N 32 N N 33 N N 34 N 8  K NL M S S S S 3 (24) Ccr EI mr K41 8 N K42 8 N K43 K44 8 8 n(t) 8 N  I A L NL A (t) P (t) K11 (An (t)) N N 0 N N 0 N N 0 N 8 n  M  0 0 0 0 Bn (t) 0 Table 5. Examined the number of the time steps (RL) N N N N N N N 8   0 0 0 0 C (t) 0 RL 250 500 750 1000 N N N N N N N 8 n Wmax (m) 0.0687 0.0686 0.0685 0.0685 0 8 N 0 8 N 0 8 N 0 8 8 n (t) 0  The number of the time steps RL = 250, the largest (20) displacement between the beams converge and the
5. Dynamic behavior of a functionally graded beam under a moving load on nonlinear viscoelastic foundation Considering Moving Mass subsequent calculations (see table 5). The numerical results are compared with the previous works to demonstrate the performance of the present study. Firstly, to further verify the present results, natural frequencies of FGMs beams composed of alumina and aluminum are calculated and compared with before study for k = 0.3 and L/h = (10, 30, 100) in table 6. The following material and beam properties: h = 0.5m. Alumina: = 380 GPa, = 3800 kg/m3, = 0.23 Aluminum: =70GPa, = 2700kg/m3, = 0.23 Table 6. Comparison of non-dimensional fundamental frequencies. BC Author L/h L/h L/h = 10 = 30 = 100 FSDBT 2.701 2.738 2.742 [7] PSDBT 2.702 2.738 2.742 FSDBT 2.774 2.813 2.817 S-S [6] PSDBT 2.695 2.737 2.742 FSDBT 2.701 2.738 2.742 Article PSDBT 2.702 2.738 2.742 FSDBT 0.970 0.976 0.977 [7] PSDBT 0.970 0.976 0.977 FSDBT 0.996 1.003 1.003 C-F [6] PSDBT 0.969 0.976 0.977 FSDBT 0.970 0.976 0.977 Fig. 2. The vertical displacements of beam (k = 0) Article PSDBT 0.970 0.976 0.977 Survey beams (S-S) boundary conditions with the FSDBT 5.875 6.177 6.214 following parameters for k = 0, L = b = 1m, h = [0.1; 0.2; [7] PSDBT 5.881 6.177 6.214 0.5] m, E = 206 (GPa), are FSDBT 6.013 6.343 6.384 calculated and compared with those of [12], [14]. C-C [6] PSDBT 5.811 6.167 6.212 Table 7: Comparison of fundamental nondimensional FSDBT 5.881 6.177 6.214 frequency. Article PSDBT 5.886 6.177 6.214 L/h Article The comparisons show that the agreement between [14] [12] FSDBT PSDBT the present results and those of [6], [7] is satisfactory. The dynamic deflections under to moving load with 0 0 7.412 7.449 7.480 7.488 time t are presented by the folowing property of beam, 10 0 8.010 8.043 8.071 8.078 100 0 12.11 12.12 12.14 12.14 load and foundation in [17]. Chart of the beam 2 displacement over time (see Fig. 2) relatively consistent 0 1 12.01 - 12.04 12.04 with the solution [17]. The article applies Timshenko 10 1 12.38 - 12.41 12.41 beam theory with the generalized coordinate method 100 1 15.32 - 15.33 15.33 while the result [17] applies Euler-Bernoulli beam theory 0 0 9.27 9.286 9.296 9.297 with the Galerkin and Runge-Kutta methods. So the 10 0 9.78 9.796 9.806 9.806 100 0 13.54 13.55 13.56 13.56 displacement line in 1/3 of the time at the beginning and 5 1/3 of the last time, there is a difference but not 0 1 13.45 - 13.46 13.46 significant and it is acceptable. Relatively small 10 1 13.80 - 13.82 13.82 displacement value less dangerous for the structure. 100 1 16.68 - 16.69 16.69
6. Huynh Van Quang The comparison is provided in Table 7. It is found that stiffening of the large beam so that the displacement of the present results are in good agreement. the beam decreases. The dynamic deflections subjected to moving mass Fig. 5. When the material distribution coefficient k are presented by the folowing formula in [9] increases, the material moves from ceramic to metal. The ceramic has a large hardness, but the metal has a small hardness, so the displacement of the beam increases. The larger the volume load is proportional to the displacement of the beam, consistent physical properties see Fig. 6. The larger the mass ratio, the more the displacement of the beam will increase. Besides, in the case of beam displacements under a moving mass larger than the case of beam moving load see Fig. 7. Fig. 3. The vertical displacements under the moving mass for full metal beam with time t. The comparisons show that the present dynamic displacements are in good agreement with the result [9] for mr = 0.1 see fig. 3. Fig. 4. When the foundation coefficient is increased, the stiffness of the foundation increases, leading to a Fig. 4. The vertical displacements of FGMs beams (S-S) under the moving mass with difference parameters of foundation K1, K2, K3, .
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