Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 1, pp. 151-166, Warsaw 2015 DOI: 10.15632/jtam-pl.53.1.151 NONLINEAR DYNAMIC ANALYSIS OF A RECTANGULAR PLATE SUBJECTED TO ACCELERATED/DECELERATED MOVING LOAD Ahmad Mamandi Department of Mechanical Engineering, Parand Branch, Islamic Azad University, Tehran, Iran e-mail: am 2001h@yahoo.com Ruhollah Mohsenzadeh DepartmentofMechanicalandAerospaceEngineering, Science and Research Branch, Islamic AzadUniversity, Tehran, Iran Mohammad H. Kargarnovin Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran In this paper, nonlinear dynamical behavior of a rectangular plate traveled by a moving mass as well as an equivalent concentrated force with non-constant velocity is studied. The nonlinear governing coupled partial differential equations (PDEs) of motion are derived by energy method using Hamilton’s principle based on the large deflection theory in conjunc- ture with the von-Karman strain-displacement relations. ThenGalerkin’smethod is used to transform the equations of motion into a set of three coupled nonlinear ordinary differential equations (ODEs)which then is solved in a semi-analyticalway to get the dynamical respon- se of the plate. Also, by using theFinite ElementMethod (FEM)withANSYS software, the obtained results in nonlinear form are verified by FEM results. Then, a parametric study is conducted by changing the size of moving mass/force and the velocity of the traveling mass/forcewith a constant acceleration/deceleration, and the outcome nonlinear results are compared to the results from linear solution. Keywords: nonlinear dynamic, plate, moving load, accelerated/deceleratedmotion 1. Introduction The theoretical and experimental investigation of the dynamic behavior of structural elements such as string, beams, plates and shells under influence of moving loads have been the subject of study of many researchers since last decades. Structures subjected to moving loads are often encountered in engineering practice. Some examples of such applications can be addressed as ropes of transporting systems, weapon firing barrels, overhead cranes, machining operations in milling and turning and plates used in road or airfield traveled by moving ground vehicles or aero vehicles, respectively. These structures may be exposed tomuch larger displacements than when they are under static loads. For small oscillations, the response of deformable bodies like plates can be suitably explained by linear equations. If the amplitude of oscillation increases then nonlinear effects appear. The cause of nonlinearitymay bematerial, geometric and inertial in nature. For such cases, certainly accurate calculation of the dynamic response is necessary for reliable design and hence better performance.Most recent articles in this topic deal with the linear behavior of plates subjected tomovingmasses and forces whereas in reality such systems naturally have nonlinear behavior. Many simple moving load problems and their analytical solutions were studied by Fryba (1999). The extensive studies of nonlinear dynamical characteristics of thin or thick beams un- der moving masses/forces were investigated in several works (Mamandi et al., 2010a,b, 2013; Mamandi and Kargarnovin, 2011a,b, 2013, 2014). A procedure incorporating the finite strip 152 A.Mamandi et al. method together with a spring system was developed and applied to treat the response of rec- tangular plate structures resting on an elastic foundation due to moving accelerated loads by Huang and Thambiratnam (2001). The effect of initial moving velocity, acceleration and initial load position on the response was discussed. Wu (2003) studied dynamic response of a rectan- gular plate subjected to multiple forces moving along a circular path. A technique to predict dynamic responses of a two-dimensional rectangular plate traveled by a transverse moving line load using the one-dimensional equivalent beammodel or the scale beams subjected to amoving concentrated load was presented by Wu (2005). The elastodynamic response of a rectangular Mindlin plate subjected to a distributedmoving mass was investigated by Gbadeyan andDada (2006). The set of governing PDEs of motion that include the effect of shear deformation and rotary inertia was expressed in a dimensionless form.A finite difference algorithmwas employed to transform the differential equations into a set of linear algebraic equations. The dynamic response of the beam-slab type bridge deck under influence ofmoving loads was obtained apply- ing the Hamilton principle with modal superposition by Law et al. (2007). Wu (2007) studied the linear dynamical response of an inclined plate subjected tomoving loads and including also the effects of the inertial, Coriolis and centrifugal forces. It has been reported that the effects of Coriolis force and centrifugal force become more significant at higher speed of the moving mass. Ghafoori et al. (2010) used a semi-analytical method to calculate the dynamic response of a rectangular plate due to amoving oscillator. The dynamic response of angle-ply laminated composite plates traversed by amovingmass or amoving forcewas investigated byGhafoori and Asghari (2010) using finite element method based on the first-order shear deformation theory. Similarly,Mohebpour et al. (2011) presentedafinite elementmodel based on thefirst order shear deformation theory to investigate the dynamic behavior of laminated composite plates traversed by a moving oscillator. A new finite element which can be used in the analysis of transverse vibrations of plates under amoving pointmasswas presented byEsen (2013). An eigen-function expansion method was employed to solve the constitutive equation of motion of a rectangular plate under various boundary conditions and the motion of a traveling mass. Vaseghi Amiri et al. (2013) studied the dynamic response of an undampedmoderately thick plate with arbitrary boundary conditions undermotion of a movingmass. The FSDT (first-order shear deformation plate theory or Mindlin plate theory) was selected as venue to derive the governing equations of motion. Combined application of the Ritzmethod, theDifferential Quadrature (DQ)method and the Integral Quadrature (IQ)method to vibration problems of rectangular plates subjected to accelerated traveling masses was investigated by Eftekhari and Jafari (2012). In the present study, the effect of geometric nonlinearity caused by stretching of the mid-plane of a rectangular plate with immovable simply supported on all edges and un- der motion of accelerated/decelerated moving mass/force on the plate dynamic respon- ses is investigated. Based on Hamilton’s principle, the nonlinear governing coupled PDEs of motion are derived and solved applying Glarkin’s method using the Adam-Bashforth- -Moulton integration method via theMATLAB solver package to obtain the dynamic response of the plate. 2. Mathematical modeling 2.1. Problem statement In Fig. 1 an isotropic and homogenous elastic rectangular plate of sides a and b (length a and width b) simply supported on all edges with density ρ, uniform thickness h, mass per unit area µ = ρh, modulus of elasticity E, Poisson’s ratio ν, bending stiffness D = Eh3/12(1−ν2) and subjected to a moving mass me with velocity V and constant acceleration/deceleration A is shown. As can be seen in Fig. 1, the origin of the Cartesian coordinate system xoy is placed Nonlinear dynamic analysis of a rectangular plate... 153 at the lower left corner of the plate. In our upcoming analysis it is assumed that the moving mass travels along a straight line at the middle of the plate width, i.e. y = b/2 (see Fig. 1). It should be mentioned that in our upcoming analysis when the moving mass enters the left side of the plate at time t =0, zero initial conditions are assumed.Moreover, in our analysis it has been assumed that the moving mass during its travel never loses its contact with the plate surface under it. In this study, the nonlinear dynamic behavior for the coupled longitudinal and transversal in-plane and out of plane displacements of a uniform rectangular plate under the action of movingmass/force is considered. It is assumed that the damping behavior follows the viscous nature. Moreover, the plate deforms within the linear elastic range and, therefore, Hook’s law is prevailing. Fig. 1. A rectangular plate of sides a and b simply supported on all edges subjected to a traveling mass me with velocity V and constant acceleration/deceleration A 2.2. Formulation According to the von-Karman nonlinear strain-displacement relations, the normal strains εx and εy and the shearing strain γxy of the middle surface for the plate shown in Fig. 1 are expressed as follows (Nayfeh andMook, 1995) εx = u,x+ 1 2 w2,x εy = v,y+ 1 2 w2,y γxy = u,y+v,x+w,xw,y (2.1) in which u(x,y,t), v(x,y,t) and w(x,y,t) represent the time dependent displacements of an arbitrarypoint located on themiddle surface of theplate in thex,y and z directions, respectively measured from the equilibrium position when unloaded. Also, in our notation the subscripts (,x), (,y) and (, t) stand for the derivative with respect to the spatial coordinates x and y and time t, respectively. To obtain the nonlinear governing differential equations of motion by applyingHamilton’s principle, the kinetic energy T of the rectangular plate under consideration is (Meirovitch, 1997; Nayfeh andMook, 1995) T = 1 2 ρh a ∫ 0 b ∫ 0 (u2,t+v 2 ,t+w 2 ,t) dxdy (2.2) and according to Kirchhoff’s plate hypothesis, the strain energy U of the plate is given by (Meirovitch, 1997) U = 1 2 a ∫ 0 b ∫ 0 h/2 ∫ −h/2 (σxεx+σyεy + τxyγxy) dxdy dz (2.3) 154 A.Mamandi et al. whereσx, σy and τxy are normal and shear in-plane stresses, respectively, and for the plate under consideration can be obtained by Hook’s law given by (Timoshenko, 1959; Ugural, 1999) σx = E 1−ν2 (εx+νεy) σy = E 1−ν2 (εy +νεx) τxy = E 2(1+ν) γxy (2.4) Now, we can establish the Lagrangian function of the system as: L = T − (U −We). Applying Hamilton’s principle on L, yields to (Meirovitch, 1997) δ t2 ∫ t1 L dt =0 ⇒ δ t2 ∫ t1 (U −T) dt = t2 ∫ t1 δWe dt (2.5) in which the total external virtual work done δWe by the gravity and the traveling mass acting on the plate at the location x = x0(t) and y = y0(t)= b/2 is (Meirovitch, 1997) δWe =− b ∫ 0 a ∫ 0 me(g+w,tt+2V w,xt+V 2w,xx+Aw,x)δw ∣ ∣ ∣ x0(t)= 1 2 At2+Vt+x0, y0(t)= b 2 dxdy (2.6) in which, mew,tt, 2meV w,xt, meV 2w,xx and Aw,x are inertial, Coriolis, centrifugal and accele- ration/deceleration induced forces acting on the elastic surface of the plate, respectively due to motion of the mass. After substitution of Eqs. (2.2), (2.3) and (2.6) into equation (2.5), performing integration and doing some mathematical simplifications one would get the nonlinear governing coupled PDEs of motion (EOMs) as follows: — the force relation in the x direction u,xx+w,xw,xx+ν(v,xy+w,yw,xy)+ 1 2 (1−ν)(u,yy+v,xy+w,xw,yy+w,yw,xy)= 1 c2p u,tt (2.7) — the force relation in the y direction u,yy+w,yw,yy+ν(u,xy+w,xw,xy)+ 1 2 (1−ν)(u,xy+v,xx+w,xw,xy+w,yw,xx)= 1 c2p v,tt (2.8) — the force relation in the z direction 1 12 h2∇4w−u,xw,xx− 1 2 w2,xw,xx−v,yw,yy − 1 2 w2,yw,yy −ν ( v,yw,xx+ 1 2 w2,yw,xx+u,xw,yy + 1 2 w2,xw,yy ) − (1−ν)(u,yw,xy+v,xw,xy+w,xw,yw,xy)+ c c2pρh w,t = 1 c2p (w,xu,tt+w,yv,tt−w,tt)− me c2pρh δ (x−x0(t))δ (y−y0(t)) · (w,xxV 2+w,tt+2w,xtV +w,xA+g) ∣ ∣ ∣ x0(t)= 1 2 At2+Vt+x0, y0(t)= b 2 (2.9) in which c2p = E/[ρ(1 − ν2)] and operator ∇4 = ( ∂4 ∂x4 + 2 ∂ 4 ∂x2∂y2 + ∂ 4 ∂y4 ) . Furthermore, δ(x −x0(t))δ(y − y0(t)) represents two dimensional Dirac’s delta function in which x0(t) and y0(t) are the instantaneous positions of the moving mass traveling on the plate. In the case the mass is traveling with velocity V and constant acceleration/deceleration A on a straight path along the trajectory parallel to the side a at themiddle of the plate width, i.e. y = b/2, then its Nonlinear dynamic analysis of a rectangular plate... 155 instantaneous position is given by x0(t)= At 2/2+V t+x0 and y0(t)= b/2, where x0 represents the initial position of the mass at the start of its motion. In addition, c (or cmn) coefficient is internal viscous damping of the plate related tomodal damping ratio, namely ζmn expressed by ζmn = cmn/[2(λmnωmn)] (Amabili, 2004), where ωmn is the natural frequency of the mth-nth mode of vibration and λmn is the modal mass of this mode given by λmn = ρhab/4 (Amabili, 2004). 3. Solution method In this study,Galerkin’smethod is chosen as a powerfulmathematical tool to analyze vibrations of the plate. Based on the separation of variables technique, the response of the plate in terms of the linear free-oscillation modes can be assumed as follows (Amabili, 2004): u(x,y,t) = m ∑ i n ∑ j pij(t)φij(x,y)= P(t)Φ T(x,y) v(x,y,t) = n ∑ k m ∑ l qkl(t)ψkl(x,y)= Q(t)Ψ T(x,y) w(x,y,t) = M ∑ v N ∑ z rvz(t)θvz(x,y)= R(t)Θ T(x,y) (3.1) where P(t), Q(t) and R(t) are vectors listing the generalized coordinate pij(t), qkl(t) and rvz(t), respectively, and φ(x,y), Ψ(x,y) and Θ(x,y) are some vectorial functions collecting the first mode shapes (eigen-functions) of φij(x,y), ψkl(x,y) and θvz(x,y), respectively. In the next step, primarilywe substituteEqs. (3.1) intoEqs. (2.7), (2.8) and (2.9), then on the resulting relations, pre-multiplying both sides of Eq. (2.7) by ΦT(x,y), Eq. (2.8) by ΨT(x,y) and Eq. (2.9) by ΘT(x,y), integrating over the interval (0,a) and (0,b) and imposing the orthogonality property of the vibrationmodes of the plate alongwith the properties of the two dimensional Dirac delta function, the resulting nonlinear coupled modal ODEs of motion in matrix form are as follows (i = l =2,4, . . . ,m, j = k =1,2, . . . ,n, v =1,2, . . . ,M, z =1,2, . . . ,N) 1 c2p m,n ∑ i,j I7,ijp̈ij(t)− m,n ∑ i,j [ I1,ij + 1 2 (1−ν)I5,ij ] pij(t)− 1 2 (1+ν) n,m ∑ k,l m,n ∑ i,j I3,klijqkl(t) − M,N ∑ v,z m,n ∑ i,j [ I2,vzij + 1 2 (1+ν)I4,vzij + 1 2 (1−ν)I6,vzij ] r2vz(t)= 0 1 c2p n,m ∑ k,l I14,klq̈kl(t)− n,m ∑ k,l [ I8,kl+ 1 2 (1−ν)I12,kl ] qkl(t)− 1 2 (1+ν) m,n ∑ i,j n,m ∑ k,l I10,ijklpij(t) − M,N ∑ v,z n,m ∑ k,l [ I9,vzkl+ 1 2 (1+ν)I11,vzkl+ 1 2 (1−ν)I13,vzkl ] r2vz(t)= 0 M,N ∑ v,z ( me c2pρh I34,vz + 1 c2p I31,vz ) r̈vz(t)+ M,N ∑ v,z ( 2meV I35,vz + cvz c2pρh I31,vz ) ṙvz(t) (3.2) + M,N ∑ v,z [h2 12 (I15,vz +2I16,vz + I17,vz)+ meV 2 c2pρh I33,vz + meA c2pρh I35,vz ] rvz(t) − M,N ∑ v,z {1 2 [I19,vz + I21,vz +ν(I23,vz + I25,vz)]+(1−ν)I28,vz } r3vz(t) 156 A.Mamandi et al. + m,n ∑ i,j M,N ∑ v,z { [I18,ijvz +νI24,ijvz+(1−ν)I26,ijvz]pij(t)+ 1 c2p I29,ijvzp̈ij(t) } rvz(t) + n,m ∑ k,l M,N ∑ v,z { [I20,klvz +νI22,klvz +(1−ν)I27,klvz]qkl(t)+ 1 c2p I30,klvzq̈rs(t) } rvz(t) = meg c2pρh M,N ∑ v,z I32,vz in which the dot mark over any parameter indicates the derivative with respect to time, i.e., t. All matrices I1 to I35 appearing in the above relations are given in Appendix A. It is clear that Eqs. (3.2) are three nonlinear coupled second-order ordinary differential equations (ODEs). The boundary conditions for a plate with immovable simple supports on all edges are (Timoshenko, 1959; Ugural, 1999): — essential BCs u = v = w =0 at x =0, x = a u = v = w =0 at y =0, y = b (3.3) — natural BCs Mx =0 → w,xx =0 at x =0, x = a My =0 → w,yy =0 at y =0, y = b (3.4) Moreover, the initial conditions (ICs) for the plate are ICs: u(x,y,0)= u,t(x,y,0)= v(x,y,0)= v,t(x,y,0)= w(x,y,0)= w,t(x,y,0)=0 (3.5) The equations of motion for a plate subjected to an equivalent concentrated moving force F ofmagnitude meg can be derived from the equations ofmotion for a plate subjected to amoving mass by neglecting the inertial effect of the travelingmass. For this system,Eqs. (2.9) and (3.2)3 are rewritten as follows (i = l =2,4, . . . ,m, j = k =1,2, . . . ,n, v =1,2, . . . ,M, z =1,2, . . . ,N) 1 12 h2∇4w−u,xw,xx− 1 2 w2,xw,xx−v,yw,yy − 1 2 w2,yw,yy −ν ( v,yw,xx+ 1 2 w2,yw,xx+u,xw,yy + 1 2 w2,xw,yy ) − (1−ν)(u,yw,xy+v,xw,xy+w,xw,yw,xy)+ c c2pρh w,t = 1 c2p (w,xu,tt+w,yv,tt−w,tt)− meg c2pρh δ (x−x0(t))δ (y−y0(t)) ∣ ∣ ∣ x0(t)= 1 2 At2+Vt+x0, y0(t)= b 2 M,N ∑ v,z 1 c2p I31,vzr̈vz(t)+ M,N ∑ v,z cvz c2pρh I31,vzṙvz(t)+ M,N ∑ v,z h2 12 (I15,vz +2I16,vz + I17,vz)rvz(t) − M,N ∑ v,z {1 2 [I19,vz + I21,vz +ν(I23,vz + I25,vz)]+(1−ν)I28,vz } r3vz(t) + m,n ∑ i,j M,N ∑ v,z { [I18,ijvz +νI24,ijvz+(1−ν)I26,ijvz]pij(t)+ 1 c2p I29,ijvzp̈ij(t) } rvz(t) + n,m ∑ k,l M,N ∑ v,z { [I20,klvz +νI22,klvz +(1−ν)I27,klvz]qkl(t)+ 1 c2p I30,klvzq̈rs(t) } rvz(t) = meg c2pρh M,N ∑ v,z I32,vz (3.6) Nonlinear dynamic analysis of a rectangular plate... 157 where Eqs. (2.7), (2.8), (3.2)1 and (3.2)2 remain unchanged. In order to solve Eqs. (3.2), all entries in the matrices listed in Appendix A should be calculated. It can be seen that the following functions (mode shapes) for φij(x,y), ψkl(x,y) and θvz(x,y) will satisfy both the linearized equations of motion and boundary conditions of the plate with immovable simpe supports on all edges (Vaseghi et al., 2013) φij(x,y)= sin iπx a sin jπy b i =2,4, . . . ,m, j =1,2, . . . ,n ψkl(x,y)= sin kπx a sin lπy b k =1,2, . . . ,n, l =2,4, . . . ,m θvz(x,y)= sin vπx a sin zπy b v =1,2, . . . ,M, z =1,2, . . . ,N (3.7) Now, we use Eq. (3.7) to calculate all matrix quantities given in Appendix A. In the next step, these evaluated matrices will be used in Eqs. (3.2), and later, the set of equations will be solved numerically using the Adams-Bashforth-Moulton integration method via MATLAB solver package to obtain values of pij(t), qkl(t) and rvz(t). By back substitution of pij(t), qkl(t) and rvz(t) intoEqs. (2.7) to (2.9), u(x,y,t), v(x,y,t) and w(x,y,t) can be obtained, respectively. Subsequently, after obtaining values for u(x,y,t), v(x,y,t) and w(x,y,t) the dynamic response of the rectangular plate under the effect of three types of massmotion: (a) accelerating, (b) de- celerating and (c) constant velocity motion are obtained. The obtained results for the plate response under each of those three types of mass/force motions are shown separately in Figs. 4 to 6. The detailed kinematical discussions of the above different motions are described below (Mamandi et al., 2010a, 2013): (a) In the case of constant accelerating type of motion (x0(t) = 0.5Aact 2 + V0t + x0, Aac = const > 0), it is assumed that the plate is at rest when the mass me enters the plate at x0 =0 and t0 =0, and with initial velocity V0 =0 and it arrives at the other end of the plate, i.e. x = a with final velocity V . The total traveling time in the plate span t1 andmass exit velocity V will be: t1 =2a/V , V = √ 2Aaca. (b) In the case of constant decelerating type of motion (x0(t) = 0.5Adet 2 + V0t + x0, Ade = const < 0), it is also assumed that the plate is at rest when themass me enters the plate at x0 = 0 and t0 = 0 and with entrance velocity V0 (non-zero initial velocity), and it stops (V = 0) at the other end of the plate, i.e. x = a. The total traveling time in the plate span t2 andmass entrance velocity V0 will be: t2 =2a/V0, V0 = √ 2|Ade|a. (c) In the case of uniform velocity type of motion (x0(t) = V t+x0), it is also assumed that the plate is at rest when the mass me enters the plate at x0 = 0 and t0 = 0 with mass constant velocity V , and it reaches the other end of the plate, i.e. x = a at the instant t3. The total traveling time in the plate span will be: t3 = a/V . 4. Verification of the results and case studies Asmentioned in the introduction, at themoment no specific results exist for the problem under consideration in the literature. Therefore, to verify the validity of the results obtained in this study,we primarily consider some special cases bywhich our results can be comparedwith those existing in the literature. 4.1. Verification of the results in linear analysis In the first attempt, we neglect the higher order terms in Eqs. (2.7) to (2.9) and structural damping for the plate, i.e., c = 0 and A = 0. This will lead us to a set of new relations for 158 A.Mamandi et al. u(x,y,t), v(x,y,t) and w(x,y,t) representing the linear form of governing EOMs of a plate subjected to amoving load with constant velocity V . By doing this, Eq. (2.9) is decoupled from Eqs. (2.7) and (2.8) and transformed into a linear form as follows (Leissa, 1969) ∇4w+ ρh D w,tt =− F D δ (x−x0(t))δ (y−y0(t)) ∣ ∣ ∣ x0(t)=V t, y0(t)= b 2 (4.1) To verify the validity of the obtained results out of our analysis, we consider a simply supported rectangular plate traveled by a moving force F with the data given by Eftekhari and Jafari (2012): a = b =10m, E =200 ·109Pa, ρ =7850kg/m3, F =3.13N, ρh/D =0.0001kg/(Nm3), y0(t) = b/2, V = 5m/s, ν = 0.3, c = 0Ns/m 2 and A = 0. By employing Eqs. (4.1) and (3.6)2 and based on the above data, the computer code was run for this case and the variation of vertical dynamic displacements w of the central point of the plate vs. time t were calculated. The outcome results are depicted and compared with other existing results in Fig. 2. A close inspection of the curves in Fig. 2 indicates very good agreements between the two outcome results. Fig. 2. Time history for the vertical displacement of the central point of the simply support plate subjected to the moving force F 4.2. Verification of the results in nonlinear analysis As described earlier, in this study, to extend checking on the validity of our obtained results we prepared appropriate APDL (ANSYS Parametric Design Language) routine in the environ- ment of ANSYS software to simulate the response of the moving force on the plate. Then, the linear and nonlinear FEM solutions have been compared with those obtained by the linear and nonlinear analytical solutions applying the mode summation technique. In the modeling of the plate, we used shell63 element defined in this software which is suitable for analyzing shell type structures. This element is a 3-D 4-noded shell element having 6 DOFs with 3-translational DOF in the x, y and z directions and 3-rotational DOF in each node about above thementioned axes. Moreover, this element is adopted to exert both in-plane and out of plane (normal) loads and suitable for large deflections. Now, to establish our calculations, we consider a plate with geometry and mechanical properties listed as: a = 4m, b = 2m, h = 0.01m, E = 200 ·109Pa, ρ =7850kg/m3, g =9.81m/s2 and ν =0.3. Figure 3 illustrates the variation of the central pointvertical deflectionw [m] of theplatewith all edges simply supported using the abovementioned data vs. time t at velocity ratio of α =1 for the traveling force of F = 0.25µgab [N] under influence of constant velocity motion using FEMand analytical analysis, respectively. From this figure, one can conclude that the results for the plate central point vertical dynamic displacement obtained byFEMand analytical solutions usingnonlinear or linear analysis are almost the samewhich showsvery good agreement between Nonlinear dynamic analysis of a rectangular plate... 159 these analytical results obtained via the mode summation technique and FEM analysis. The suitable number of elements which has been used for the plate to converge the linear/nonlinear results is 288 elementswith 24 elements used in the length a and 12 elements used in thewidth b of the plate. It should bementioned that themaximum relative difference between the obtained results for the plate central point vertical deflection using linear and FEM solutions is 4.8% at the instant 0.073s, and also themaximum relative difference between the obtained results using nonlinear analysis and FEM solution is 1% at the instant 0.09s. Fig. 3. Variation of the vertical deflection of the central point w [m] vs. time t [s] for the simply supported plate affected by the moving force F =0.25µgab [N] under constant velocity motion at α =1 using analytical and FEM analysis for linear and nonlinear solutions 5. Results and discussions After being satisfied about the validity of the solution technique, the plate central point instan- taneous dynamic vertical deflection is calculated in the next step. In obtaining these results, a steel plate with the same specifications as mentioned above in Section 4.2, but in addition having ζ =0.033 (Mamandi et al., 2010a,b) is considered. It should be mentioned that all deflection variations vs. moving mass instantaneous posi- tion are given in a non-dimensional form that is wmax/w0. Moreover, it has to be pointed out that based on the conducted convergence study related to the linear and nonlinear analyses, 9 modes of vibration are taken into account for steady state answers for u(x,y,t), v(x,y,t) and w(x,y,t). Toclarify the results and in order tohaveabetter insight into interpreting thevariation of the obtained results,we tried topresent the results indimensionless forms.So,webeginwithdefining the normalized maximum dynamic vertical deflection of the central point of the plate to its maximum static response at the same point. The static downward deflection of the plate central point under a concentrated mass applied at the same point is equal to w0 = 0.01651mega 2/D (Timoshenko, 1959). Moreover let us define the velocity ratio as α = T1/T = V/Vp in which Vp = a/T1 =(ωna)/(2π), whereT1,T andVp denote thefirstnatural period (fundamental period of transverse motion) of the plate, the total time taken by the moving load to cross from one side to the opposite side of the plate and the velocity of the reference load that would take the time of T1 to traverse the plate of length a, respectively. Moreover, ωn is the natural frequency of the plate given by (Leissa, 1969) ωn = ωij = π 2 [( i a )2 + (j b )2] √ D ρab i,j =1,2, . . . ,n It should bepointed out that in conjuncturewith the stretching effect of themid-plane of the plate, the geometric nonlinearity behavior of the plate also depends on the ratio of theweight of 160 A.Mamandi et al. the moving mass as well as amplitude of the equivalent concentrated moving force to the plate weight. For sure, we have done this study before any further calculation. It has been noticed that when this ratio is usually greater than 0.1, the nonlinear geometric effects come into play. Figures 4 and 5 show the variation of the dimensionless dynamic vertical deflection (wmax/w0) vs. dimensionless time t/ti (i =1,2,3) at the central point of the simply supported rectangular plate traversed by themovingmass of me =0.25µab [kg] and an equivalent concen- trated force of F =0.25µgab [N], respectively with different velocity ratios (α =1,2,3,4) under the influence of three types of motion using linear and nonlinear solutions, respectively. As can Fig. 4. Variation of the dimensionless dynamic vertical deflection (wmax/w0) at the central point of the plate vs. normalized time (t/ti) for the simply supported rectangular plate traversed by the moving mass of me =0.25µab [kg] with different velocity ratios (α =1,2,3,4) under influence of three types of motion; (a) acceleratedmotion, (b) deceleratedmotion, (c) uniform velocitymotion; (—) nonlinear analysis, (- - -) linear analysis be seen from Figs. 4 and 5, in the accelerating and decelerating types of motion by increasing the velocity ratio α up to α =3 and 2, respectively, the value of maximum nonlinear dynamic downward deflection increases and the reverse trend prevails afterwards, whereas in uniform ve- Nonlinear dynamic analysis of a rectangular plate... 161 Fig. 5. Variation of the dimensionless dynamic vertical deflection (wmax/w0) at the central point of the plate vs. normalized time (t/ti) for the simply supported rectangular plate traversed by the moving force of F =0.25µgab [N] with different velocity ratios (α =1,2,3,4) under influence of three types of motion; (a) acceleratedmotion, (b) deceleratedmotion, (c) uniform velocitymotion; (—) nonlinear analysis, (- - -) linear analysis locity motion the reduction trend of themaximum dynamic downward deflection is always seen no matter what type of analysis is used. It is noticed that in the accelerating type of motion, the maximum dynamic deflection is reached at a much later time than the other two cases. Moreover, in the decelerating type of motion, the range of variation of the maximum dynamic deflection is larger with respect to the other two types of motion. Also, it can be seen from Figs. 4 and 5 that for a higher velocity ratio, i.e. α =4, in uniform velocity motion, the vertical dynamic displacement of the plate central point yields a smaller value at the time of leaving the plate, whichmeans the plate does not have enough time to respond accordingly against the fast speed of the moving mass/force. In addition, from these figures, it can be seen that in the decelerating type ofmotion in lower velocity ratios, i.e. α =1 and 2, there is a reverse (upward) displacement for the central pointwhich occurs usuallywhen the load leaves the plate. It can be 162 A.Mamandi et al. Fig. 6. Variation of the dimensionless dynamic vertical deflection (wmax/w0) at the central point of the plate vs. normalized time (V t/a) for the simply supported rectangular plate traversed by the moving mass of me for different velocity ratios in the constant velocity type of motion; (a) α =0.75, (b) α =1, (c) α =1.25, (d) α =1.5 Nonlinear dynamic analysis of a rectangular plate... 163 observed in Figs. 4 and 5 that the obtained results from nonlinear solution have almost smaller values that those calculated by linear solution. In addition, from Figs. 4 and 5, it is concluded that themaximumdifference for the plate central point deflection between nonlinear and linear solutions happens primarily in the deceleratingmotion andwith a smaller difference in the case of uniform velocity and then in the accelerating type of motion, respectively. To study the effect of weight of themovingmass to generate the nonlinearity behavior of the plate, Fig. 6 shows the variation of the dimensionless dynamic vertical deflection (wmax/w0) of the simply supported rectangular plate for different velocity ratios (α = 0.75, 1, 1.25 and 1.5) traversed by differentmovingmasses me (me = cmµab [kg], cm =0.5, 1, 2, 3 and 4) vs. the nor- malized instantaneous mass position, i.e. V t/a using nonlinear analysis in the constant velocity type of motion. It can be seen from this figure that when α increases up to the velocity ratio α =1.25, themaximumvalue of instantaneous dynamic deflection increases and decreases after- wards. The maximum dynamic deflection for all cases in this figure occurs at me =0.5µab [kg] at α =1.25. 6. Conclusions Threenonlinear coupledpartial differential equations ofmotion for the in-plane andout-of-plane displacements of a rectangular plate subjected to an accelerated/decelerated traveling mass as well as an equivalent concentrated force are solved, and the results are following: • It can be seen that in the accelerating type of motion, the maximum dynamic deflection is reached at a much later time than in the other two types of motion. • It is concluded that in the decelerating type of motion, the range of variation of the maximum dynamic deflection is larger with respect to the other two types of motion. • Forhighervelocity ratios in theuniformvelocitymotion, thevertical dynamicdisplacement of the plate central point yields a smaller value at the time of leaving the plate. • It is seen that in the moving mass/force problem in three types of motion, the obtained results by nonlinear solution have almost smaller values than those calculated by linear solution. • It is observed that in conjuncture with the stretching effect of the mid-plane of the plate when the ratio of theweight of themovingmass as well as the amplitude of the equivalent concentrated moving force to the plate weight is usually greater than 0.1, the geometric nonlinearity behavior of the plate comes into play. • It is concluded that themaximumdifference for the plate central point deflection between nonlinear and linear solutions happens primarily in the decelerating motion and with a smaller difference in the case of uniform velocity and then in the accelerating type of motion, respectively. Appendix A The definition of differentmatrices used in calculation of the nonlinear coupled ODEs of modal relations Eqs. (3.2) are I1,ij = b ∫ 0 a ∫ 0 d2φij(x,y) dx2 φij(x,y) dxdy I2,vzij = b ∫ 0 a ∫ 0 dθvz(x,y) dx d2θvz(x,y) dx2 φij(x,y) dxdy I3,klij = b ∫ 0 a ∫ 0 d2ψkl(x,y) dxdy φij(x,y) dxdy I4,vzij = b ∫ 0 a ∫ 0 dθvz(x,y) dy d2θvz(x,y) dxdy φij(x,y) dxdy 164 A.Mamandi et al. I5,ij = b ∫ 0 a ∫ 0 d2φij(x,y) dy2 φij(x,y) dxdy I6,vzij = b ∫ 0 a ∫ 0 dθvz(x,y) dx d2θvz(x,y) dy2 φij(x,y) dxdy I7,ij = b ∫ 0 a ∫ 0 φ2ij(x,y) dxdy I8,kl = b ∫ 0 a ∫ 0 d2ψkl(x,y) dy2 ψkl(x,y) dxdy I9,vzkl = b ∫ 0 a ∫ 0 dθvz(x,y) dy d2θvz(x,y) dy2 ψkl(x,y) dxdy I10,ijkl = b ∫ 0 a ∫ 0 d2φij(x,y) dxdy ψkl(x,y) dxdy I11,vzkl = b ∫ 0 a ∫ 0 dθvz(x,y) dx d2θvz(x,y) dxdy ψkl(x,y) dxdy I12,kl = b ∫ 0 a ∫ 0 d2ψkl(x,y) dx2 ψkl(x,y) dxdy I13,vzkl = b ∫ 0 a ∫ 0 dθvz(x,y) dy d2θvz(x,y) dx2 ψkl(x,y) dxdy I14,vzkl = b ∫ 0 a ∫ 0 ψ2kl(x,y) dxdy I15,vz = b ∫ 0 a ∫ 0 d4θvz(x,y) dx4 θvz(x,y) dxdy I16,vz = b ∫ 0 a ∫ 0 d4θvz(x,y) dx2dy2 θvz(x,y) dxdy I17,vz = b ∫ 0 a ∫ 0 d4θvz(x,y) dy4 θvz(x,y) dxdy I18,ijvz = b ∫ 0 a ∫ 0 dφij(x,y) dx d2θvz(x,y) dx2 θvz(x,y) dxdy I19,vz = b ∫ 0 a ∫ 0 (dθvz(x,y) dx )2d2θvz(x,y) dx2 θvz(x,y) dxdy I20,klvz = b ∫ 0 a ∫ 0 dψkl(x,y) dy d2θvz(x,y) dy2 θvz(x,y) dxdy I21,vz = b ∫ 0 a ∫ 0 (dθvz(x,y) dy )2d2θvz(x,y) dy2 θvz(x,y) dxdy I22,klvz = b ∫ 0 a ∫ 0 dψkl(x,y) dy d2θvz(x,y) dx2 θvz(x,y) dxdy I23,vz = b ∫ 0 a ∫ 0 (dθvz(x,y) dy )2d2θvz(x,y) dx2 θvz(x,y) dxdy I24,ijvz = b ∫ 0 a ∫ 0 dφij(x,y) dx d2θvz(x,y) dy2 θvz(x,y) dxdy I25,vz = b ∫ 0 a ∫ 0 (dθvz(x,y) dx )2d2θvz(x,y) dy2 θvz(x,y) dxdy I26,ijvz = b ∫ 0 a ∫ 0 dφij(x,y) dy d2θvz(x,y) dxdy θvz(x,y) dxdy I27,klvz = b ∫ 0 a ∫ 0 dψkl(x,y) dx d2θvz(x,y) dxdy θvz(x,y) dxdy Nonlinear dynamic analysis of a rectangular plate... 165 I28,vz = b ∫ 0 a ∫ 0 dθvz(x,y) dx dθvz(x,y) dy d2θvz(x,y) dxdy θvz(x,y) dxdy I29,ijvz = b ∫ 0 a ∫ 0 dθvz(x,y) dx φij(x,y)θvz(x,y) dxdy I30,klvz = b ∫ 0 a ∫ 0 dθvz(x,y) dy ψkl(x,y)θvz(x,y) dxdy I31,vz = b ∫ 0 a ∫ 0 θ2vz(x,y) dxdy I32,vz = θvz(x = x0(t),y = b/2) I33,vz = d2θvz(x = x0(t),y = b/2) dx2 θvz(x = x0(t),y = b/2) I34,vz = θ 2 vz(x = x0(t),y = b/2) I35,vz = dθvz(x = x0(t),y = b/2) dx θvz(x = x0(t),y = b/2) Acknowledgment Thispaper is dedicated toAhmadMamandi’s dearestProfessorDr.MohammadHosseinKargarnovin, who passed away on Monday, November 4, 2013. 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