On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method

Transcription

1 Acta Mech. Sin. (211) 28(1): DOI 1.17/s y RESEARCH PAPER On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method Keivan Kiani Ali Nikkhoo Received: 31 October 21 / Revised: 11 June 211 / Accepted: 11 June 211 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 212 Abstract This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of motion are developed based on Lagrange s equations via reproducing kernel particle method (RKPM). For a particular case of a simply supported beam, Galerkin method is also employed to verify the results obtained by RKPM, and a reasonably good agreement is achieved. Variations of the maximum dynamic deflection and bending moment associated with the linear and nonlinear beam theories are investigated in terms of moving mass weight and velocity for various beam boundary conditions. It is demonstrated that for majority of the moving mass velocities, the differences between the results of linear and nonlinear analyses become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. Except for the cantilever beam, the nonlinear beam theory predicts higher possibility of moving mass separation from the base beam compared to the linear one. Furthermore, the accuracy levels of the linear beam theory are determined for thin beams under large deflections and small rotations as a function of moving mass weight and velocity in various boundary conditions. Keywords Nonlinear beam theory Moving mass-beam interaction Euler Bernoulli beam theory Reproducing kernel particle method (RKPM) Galerkin method (GM) K. Kiani Department of Civil Engineering, Islamic Azad University, Chalous Branch, Chalous, Iran A. Nikkhoo ( fi ) Department of Civil Engineering, University of Science and Culture, P.O. Box , Tehran, Iran nikkhoo@usc.ac.ir 1 Introduction The beam is the most common structural element in the modeling of various structures widely used in aerospace, mechanical and civil engineering sciences. The small deflection theory for beams (linear beam theory) is commonly hypothesized for evaluating the dynamic response of such structures under the applied moving loads. Nevertheless, in assessing of large amplitude vibrations of beams under large moving loads, the nonlinear terms in the governing equations of motion could not be ignored. Hence, the results of linear beam theory can not be used for design purposes, since the structure experiences large deflections and rotations. On the other hand, incorporating the inertial effects of the moving load into the problem formulation-known as the moving mass problem-would generally intensify the dynamic response of the structure. Therefore, the nonlinear analysis is necessary in practical applications of beam-like structures subjected to moving masses, particularly for relatively slender beams, which usually undergo large deflections. A large body of studies on the problems of various structures with linear behavior traversed by moving systems has been conducted by Frýba [1]. Concerning the problem of moving loads acted on thin beams with geometric nonlinearity, Hino et al. [2] analyzed the nonlinear vibration of immovably supported beams by using a Galerkin finite element method. The results obtained by using the assumed nonlinear models were compared to those of the linear model. Yoshimura et al. [3] studied the dynamic motion and large deflections of simply supported Euler Bernoulli beams utilizing Galerkin method of weighted residuals. They showed that only a few terms of the Galerkin series would be needed for proper convergence of the problem. Moreover, it was demonstrated that the differential equations derived from the proposed method could be solved by the Newmark-β scheme in the time domain which was proved to be numerically stable. In another work, Wang and Chou [4] employed the large

2 On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method 165 deflection theory to derive the equations of motion of Timoshenko beams for studying the coupling effect of an external force with the weight of the beam. The dynamic response of the beam was obtained via Galerkin method. Their work was mainly focused on the effect of weight of the beam on its dynamic deflection and bending moment predicted by small and large deflection theories. According to the inherent complexities of the moving mass problems in the linear beams, some studies have been carried out using different numerical schemes, e.g., [5 1]. As a nonlinear moving mass problem, Xu et al. [11] derived the nonlinear equations of motion for a finite elastic beam traversed by a moving mass. A finite difference method (FDM) combined with the perturbation technique was employed to solve the resulting boundary value problem. Their results for a simply supported beam indicated that there would be some differences between the moving force and moving mass dynamic deflections. It was also shown that the effect of the friction force between the mass and the beam on the longitudinal motion could be significant. Siddiqui et al. [12] investigated the motion of a flexible cantilever beam carrying a moving spring-mass system via the Rayleigh Ritz method as well as perturbation method of multiple scales. The coupled partial differentialequationsdescribing the system motion were derived for an Euler Bernoulli beam. They showed that there would be a good agreement between the results of the perturbation solution and those of the numerical approach. Besides, it was highlighted that the interaction between the mass and the beam leads to nonlinearity effects which make the system exhibit internal resonance. According to the literature, numerical solutions for the problem of nonlinear beams subjected to the moving mass were limited to the mesh-based methods and FDM. An alternative to the traditional element-based approaches such as the finite element method (FEM) is the class of meshless methods based on reproducing kernels. One of the most famous methods in this family is the reproducing kernel particle method (RKPM). Liu et al. [13 15] have developed the RKPM and demonstrated its applications ranging from structural dynamics to large deformation problems. This new methodology, employs a grid of particles, eliminating the need for a mesh and some difficulties associated with the FEM such as mesh entanglement in large deformation problems. Additionally, RKPM employs shape functions that allow the interpolation of field variables to be achieved at the required level. The new shape functions are constructed based on window functions; higher continuity of window functions would lead to higher continuous shape functions. Therefore, by utilizing the appropriate window and shape functions, the field variables of the problem are reproduced more smoothly with higher continuity in comparison to those of FEM, and the accuracy of the solution is thus improved. In this regard, RKPM could be exploited as a suitable numerical method for model refinement, fracture problems, composite problems and large deformation problems. The application of RKPM to the problems of structural dynamics was performed by Liu et al. [16]. Moreover, its application to elastic and elastic-plastic one-dimensional bar problems for both small and large deformations as well as two dimensional ones, showed remarkable results in comparison to the results of FEM [13]. Recently, Kiani et al. [8] explored the capabilities of various beam models in capturing the dynamic response of beam-like structures acted upon by a moving mass using RKPM. The influences of mass weight and velocity of the moving mass as well as slenderness of the beam on its maximum dynamic deflection and bending moment are scrutinized based on the classical and shear deformable beam models. In another work, Kiani [17] investigated free transverse vibration of embedded single-walled nanotubes with various boundary conditions via RKPM. For this purpose, the nanotube with elastically attached springs was modeled according to the nonlocal Euler Bernoulli, Timoshenko, and higher-order beam theories. The obtained results for the first five frequencies of the nanotube by RKPM were compared with those of other works and a good agreement was achieved. In this study, the discrete equations of motion for an elastically supported Euler Bernoulli beam under the excitation of a moving mass are developed based on Lagrange equations. For this purpose, RKPM is employed for spatial discretization of the problem and the Crank Nicholson scheme is employed for time discretization of the ordinary differential equations of motion. The validity of the proposed numerical solution is then examined based on the obtained results by the Galerkin method. The maximum values of deflection and bending moment are considered as the crucial design parameters. The major contribution of this work rests on the variation of design parameters associated with the linear and nonlinear beam theories in terms of moving mass weight and velocity for various boundary conditions. Moreover, the possibility of mass separation from the base beam is studied and discussed based on the linear and nonlinear beam analyses. Finally, the accuracy levels of the linear beam theory for geometrically nonlinear behavior of beams as a function of moving mass weight and velocity are determined for different beam boundary conditions. 2Definition of the mechanical problem Consider a thin beam of length l b, which is constrained at its ends by transverse and torsion springs with constants K z and K y, as shown in Fig. 1. The beam is axially fixed at its left end, and is acted upon by a moving mass of mass M, and velocity v along the beam. A Cartesian coordinate system xyz is assumed to be fixed to the left end of the undeformed beam with the x-axis coincident with its neutral axis, and the z-axis pointing towards the applied gravitational acceleration, g. Let U x (x, z, t) andu z (x, z, t) signify the longitudinal and transverse displacement components of

3 166 K. Kiani, A. Nikkhoo the beam while E xx /E xz and S xx /S xz present the longitudinal/transverse Green Lagrange strains and stresses, respectively. The following assumptions are made in modeling the mechanical problem. First, the material of the beam is linear homogeneous isotropic with the elastic modulus of E b,the constant cross-section area of A b, and the uniform density of ρ b. Second, the excited beam could experience large deformation with small rotation. Third, the moving mass travels with a constant velocity along the beam. From the previous assumption, the magnitude of the transverse component of velocity would be negligible. Fourth, the moving mass material is a rigid medium, and it would be in contact with the base beam during excitation. As a result, from the second and third assumptions, the transverse acceleration of the moving mass is ( D 2 U z Ü zm (Ü Dt )(x 2 z + 2v U z,x + v 2 U z,xx ) (xm,z M ), (1) M,z M ) in which D/Dt is the material derivative and (x M, z M )denotes the contact point location of the moving mass on the top surface of the beam. Therefore, the applied contact force at the mentioned point is expressed as F c M(g Ü zm )δ(x x M )δ(z z M )H(l b x M ), (2) where δ and H are the Dirac delta and Heaviside step functions, respectively. It is worth mentioning that the possibility of moving mass separation from the base beam is monitored by the sign of the applied contact force. The onset of separation occurs just as the sign of F c changes from positive to negative. Lee [5] examined the effect of separation between the moving mass and the base beam on the dynamic response of the structure. The obtained results showed that this phenomenon would have significant effects on the vibration of the structure, especially for higher levels of mass weight and velocity of the moving mass. It is emphasized herein that in all analysis of this article, moving mass would be in contact with the beam at all times; however, the possibility of the separation of moving mass from the base beam in the context of large deflection is of particular interest. Fig. 1 An elastic Euler Bernoulli beam constrained at its ends by elastic springs under excitation of a moving mass 3 Theoretical formulations According to the Euler Bernoulli beam theory, the displacement components in the plane of the beam loading could be written as U x (x, z, t) u(x, t) zw,x (x, t), (3) U z (x, z, t) w(x, t), in which u(x, t) represents the longitudinal displacement of the neutral axis of the beam. Therefore, the components of the Green Lagrange strain field are obtained as E xx U x,x (U2 x,x + U 2 z,x) u,x (u2,x + w 2,x) zw,xx (1 + u,x ) z2 w 2,xx, (4) E xz 1 2 (U x,z + U z,x + U x,x U x,z + U z,x U z,z ) 1 2 w,x(zw,xx u,x ), according to the second assumption, the terms in z 2 as well as the products of w,x with the derivatives of the displacements are ignored in the numerical analyses [18]. Subsequently, the only nonzero component of the strain fieldwouldbe E xx u,x (u2,x + w 2,x) zw,xx, (5) basedonthefirst assumption, the constitutive relation may be stated by a linear relation between the Green Lagrange strain and stress as S xx E b E xx. Hence, the axial force and bending moment in the beam is provided by N b S xx da E b A b [u,x + 1 ] A 2 (u2,x + w 2,x), (6) M b zs xx da E b I b w,xx. A The Lagrangian functional is expressed as L T (U + V), (7)

4 On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method 167 where L is the total energy, T is the kinetic energy, U is the strain energy, and V is the potential energy of the beam and springs system under a moving mass loading. Employing the penalty method to impose the essential boundary conditions of the longitudinal displacement, the components of L are outlined as follows T 1 ρ b ( U x 2 + U z 2 )dω 2 Ω 1 ρ b [A b ( u 2 + ẇ 2 ) + I b ẇ 2 2,x]dx, (8) U 1 σ xx ɛ xx dω + 1 (α u u 2 + K y w 2,x + K z w 2 )dγ 2 Ω 2 Γ b 1 [ {E b A b u,x + 1 ] 2 } 2 2 (u2,x + w 2,x) + Eb I b w 2,xx dx + 1 (α u u 2 + K y w 2,x + K z w 2 )dγ, (9) 2 Γ b V M(g Ü zm )U z δ(x x M )δ(z z M )H(l b x M )dγ Γ b M[g (ẅ + 2vẇ,x + v 2 w,xx )] wδ(x x M )H(l b x M )dx, (1) in which Ω is the beam domain, Γ b denotes the boundary of the beam domain, and α u represents the penalty parameter associated with the longitudinal degree of freedom (DOF) which is set equal to 1 6 E b A b in this study. The only unknown variables of the problem are discretized as u(x, t) φ u I (x)u I(t); w(x, t) φ w I (x)w I(t); I 1, 2,, N P, where I is a free index, and N P is the number of particles distributed in the spatial domain of the beam. The dual parameters (φ u I,φw I )and(u I, w I ) are, respectively, the RKPM shape functions and the nodal parameter values associated with the DOFs of the I-th particle. Furthermore, the Lagrange s equations are expressed as L d L, u I dt u I L d I 1, 2,, N P, (11) L, w I dt ẇ I by substituting the equivalent values of u(x, t) andw(x, t) in terms of u I (t) andw I (t) into Eqs. (8) (1) and utilizing the Lagrange equations (see Appendix (1)), the coupled discrete equations of motion for the mechanical problem are obtained as and ρ b A b φ u I üdx + E b A b [u,x + 1 ] 2 (u2,x + w 2,x) (1 + u,x )φ u I,x Γ dx + α u uφ u I dγ, (12) b ρ b (A b ẅφ w I + I b ẅ,x φ w I,x )dx [ + {E b A b u,x + 1 ] } 2 (u2,x + w 2,x) w,x φ w I,x dx + (K z φ w I w + K yφ w I,x w,x)dγ Γ b + Mφ w I (ẅ + 2vẇ,x + v 2 w,xx )δ(x x M )H(l b x M )dx Mgφ w I δ(x x M)H(l b x M )dx. (13) The recent equations could be rewritten in the matrix form as M ẍ f ext f int, (14) or M uu M uw ü M wu M ww ẅ f ext u f ext f int u w f int, (15) w in which the non-zero submatrices are defined as follows M uu IJ M ww IJ ρ b A b φ u I φu Jdx, (16) ρ b (A b φ w I φw J + I bφ w I,x φw J,x )dx +Mφ w I (x M)φ w J (x M)H(l b x M ), (17) { f ext w } I Mgφ w I (x M)H(l b x M ), (18) { f int u } I E b A b [u,x + 1 ] 2 (u2,x + w 2,x) (1 + u,x )φ u I,x dx + α u uφ u I dγ, (19) Γ b [ { f int w } I {E b A b u,x + 1 ] 2 (u2,x + w 2,x) w,x φ w I,x } +E b I b w,xx φ w I,xx dx + M(v 2 w,xx + 2vẇ,x ) φ w I δ(x x M)H(l b x M )dx + (K z wφ w I + K y w,x φ w I,x )dγ, (2) Γ b where f ext and f int are defined as the external and internal force vectors, respectively. It should be noted that f int is a nonlinear vector function in terms of the unknown vector x. Moreover, it is time dependent. Therefore, an appropriate method should be employed to obtain a proper response at each time step. To this end, a new vector y is defined as y ẋ. Hence, one could arrive at ˆM ż ˆf, (21) where y z x, M ˆM I, f ext f int ˆf y, (22)

5 168 K. Kiani, A. Nikkhoo in which I represents the identity matrix. In order to discretize Eq. (21) in the time domain, the finite difference approximation method is implemented as ż (z i+1 z i )/Δt. Therefore, Mz i+1 Mz i Δt f, (23) in which M (1 θ) ˆM i + θ ˆM i+1, (24) f (1 θ) ˆf i + θ ˆf i+1, where Δt t i+1 t i,[] i represents the value of the parameter [ ] at the time t i,andθis the weight parameter of time such that θ 1. The value of this parameter is set equal to.5 throughout this paper (i.e., central difference method or Crank Nicholson scheme). Utilizing Newton s method to calculate z i+1 in Eq. (23) at each time step, one could conclude KΔz i+1 f, (25) in which Δz i+1 z new i+1 zold i+1, ( ˆf ) K M Δtθ z zz old i+1, (26) f Mz old i+1 + Mz i +Δt f, where z old i+1 is the previous value of z i+1 in the iteration process. The elements of the matrix ˆf / z are evaluated as presented in Appendix (2). The unknown parameters of Eq. (25) could be determined by executing iterative operations until accurate results for z new i+1 are obtained at each time step. In this work, the convergence criterion is set to be Δz i+1 / z old i+1 < Results and discussions 4.1 Comparison of the RKPM results with those of Galerkin method In this part, the predicted results of nonlinear RKPM are compared with the obtained results by the Galerkin method (GM) for a simply supported thin beam (see Appendix (3)). In all the analyses using RKPM, 15 uniformly distributed particles and 6 Gaussian points are adopted for each computational cell. The particle shape functions are generated for the dilation parameter value of 3.5, the linear base function and the exponential window function. Additionally, the nondimensional slenderness, velocity and mass parameters are assumed as λ l b /r, V N v/v [7], and M N M/ρ b A b l b, correspondingly, where r I b /A b and v π/l b Eb I b /ρ b A b. The geometric and material properties of the beam with a rectangular cross section having unit width are considered as: l b 1 m, E b 21 GPa, λ 8, and ρ b 7 85 kg m 3. Moreover, K z 1 6 E b I b /l 3 b and K y 1 3 E b I b /l b are considered for the numerical analyses in the case of undeflected and unrotated ends (i.e., w, w,x ), respectively. Additionally, the values of the time step are set equal to l b /(3v) andl b /(3v ) in the time intervals of t l b /v, andt > l b /v, correspondingly. In the case of GM analyses, the first 11 vibration modes of the beam have been considered. Based on the nonlinear RKPM and GM analyses, the time history of the normalized deflection and bending moment of the midspan point of a simply supported-pinned movable (SS-PM) beam have been plotted in Figs. 2a and 2b, respectively. Moreover, the predicted values of normalized contact force (F c,n F c /Mg) at the contact point Fig. 2 Comparison of the obtained results by RKPM with those by RRM for an SS-PM thin beam according to the nonlinear analyses. a Normalized dynamic deflection of the midspan point; b Normalized dynamic bending moment of the midspan point; c Normalized contact force at the contact point of the moving mass and the thin beam. V N.75, M N.4, λ 8)

6 On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method 169 (x x M ) by RKPM and GM are presented in Fig. 2c. As can be seen, there is a good agreement between the predicted results of RKPM and those of GM over the whole range of times. On the other hand, both methodologies show the sudden change of the contact force sign from positive to negative shortly before the departure of the moving mass from the beam end. Therefore, the maximum negative contact force could be introduced as a rational criterion for the possibility of the separation of moving mass from the base beam. This issue will be addressed in some detail in the next part for thin beams with different boundary conditions. It should be noted herein that the application of GM to other boundary conditions excluding the SS case, would result in a rather complicated manipulation process in evaluating the integrals. As a result, RKPM is utilized to explore the effects of moving mass weight and velocity on the linear and nonlinear responses of the thin beams with various boundary conditions. 4.2 Parametric studies for different boundary conditions For investigating the effects of different parameters on the maximum deflection and bending moment of the beam, two normalized design parameters W max,n W max,dyn /W max,st and M max,n M max,dyn /M max,st are considered where W max,dyn and M max,dyn denote the maximum dynamic deflection and the maximum bending moment along the beam caused by the moving mass, respectively. W max,st and M max,st are the appropriate maximum static deflection and bending moment along the Euler Bernoulli beam under statically applied point load Mg. A proper static analysis would result in Mgl 3 b /48E bi b, Mgl 3 b /192E bi b, Mgl 3 b /E bi b,and Mgl 3 b /3E bi b for W max,st corresponding to simple-simple (SS), clamped-clamped (CC), simple-clamped (SC) and clampedfree (CF) boundary conditions, respectively. Similarly, the appropriate values of M max,st for SS, CC, SC and CF boundary conditions are Mgl b /4, Mgl b /8,.174Mgl b,andmgl b, respectively. In Fig. 3, time history of the normalized deflection of an SS beam at its midspan is depicted for different moving mass velocities. As is observed, for low levels of moving mass velocity (Fig. 3a), the results of linear and nonlinear beam theories are almost the same. By increasing the moving mass velocity, a larger distinction between the results of two theories could be seen. To examine the role of different parameters in affecting the dynamic behavior, the variations of normalized design parameters based on both linear and nonlinear analyses in terms of M N and V N are depicted in Figs. 4 7 for different boundary conditions. As it a general result, in the linear analysis, there would be a roughly linear relation between the values of design parameters and M N, except for the CF boundary condition [8]. It is also indicated that for V N.5, the higher the moving mass weight is, the higher the design parameters values (see Figs. 4 6) are. As it may be noticed in Fig. 7, for the cantilever beam, higher values of M N would result in lower values of design parameters regardless of the moving mass velocity. Concerning the nonlinear analysis results, the values of design parameters are lower than those associated with the linear analysis in most of the studied cases. Furthermore, for the majority of moving mass weights, the differences between the linear and nonlinear results become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. It could be interesting to note that for an SS beam, the linear analysis would underestimate the values of M max,n for high moving mass weights and velocities. Fig. 3 Normalized dynamic deflection of the linear and nonlinear analyses at the midspan of an SS beam. a V N.1; b V N.25; c V N.5; d V N.75. M N.3, τ vt/l b )

7 17 K. Kiani, A. Nikkhoo Fig. 4 Effect of the moving mass weight on the maximum linear and nonlinear dynamic response of an SS beam for various values of the moving mass velocity. a Normalized maximum deflection vs. normalized moving mass weight; b Normalized maximum bending moment vs. normalized moving mass weight Fig. 5 Effect of the moving mass weight on the maximum linear and nonlinear dynamic response of a CC beam for various values of the moving mass velocity. a Normalized maximum deflection vs. normalized moving mass weight; b Normalized maximum bending moment vs. normalized moving mass weight Although the mathematical model involves a basic assumption that the mass would always be in contact with the base beam, it would be worthwhile to investigate the possibility of mass separation from the base beam during excitation. This issue is studied for different beam boundary conditions as shown in Figs. 8a 11a. To this end, the variation of normalized minimum contact force (F min,n F c,min /Mg) in terms of moving mass weight and velocity are shown for both linear and nonlinear analyses. As is obvious, except for the CF boundary condition, the nonlinear beam theory predicts a higher possibility of mass separation in comparison to the linear one. Moreover, the magnitude of the nor-

8 On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method 171 malized minimum and maximum contact forces severely increase for higher levels of moving mass weight and velocity, exclusively in the nonlinear analysis. It implies that the differences between the linear and nonlinear analyses results increase as the moving mass weights and velocities increase. It was also declared by Lee [5] that the separation between the moving mass and the base beam could have significant effect on the dynamic response of the beam, particularly for high levels of mass weight and velocity of the moving mass. As seen in Figs. 8a 11a, for most of the considered velocity of the moving mass, the highest possibility of mass separation occurs in the case of CF boundary condition while the lowest possibility is observed in the case of CC boundary condition. Fig. 6 Effect of the moving mass weight on the maximum linear and nonlinear dynamic response of an SC beam for various values of the moving mass velocity. a Normalized maximum deflection vs. normalized moving mass weight; b Normalized maximum bending moment vs. normalized moving mass weight Fig. 7 Effect of the moving mass weight on the maximum linear and nonlinear dynamic response of a CF beam for various values of the moving mass velocity. a Normalized maximum deflection vs. normalized moving mass weight; b Normalized maximum bending moment vs. normalized moving mass weight

9 172 K. Kiani, A. Nikkhoo Equally important is the effectof moving mass velocity on the values of design parameters, as shown in Figs for various boundary conditions and for both the linear and nonlinear beam theories. As expected, in most of the cases, a linear beam theory provides an upper bound for nonlinear beam theory regardless of the beam boundary condition and the moving mass weight and velocity. An appreciable distinction between the results of the linear and nonlinear beam theories initiates at certain value of moving mass velocity, which varies with the beam boundary condition. For example, for M N.2, the above-mentioned normalized velocities for CF, SS, SC and CC boundary conditions are.15,.2,.3 and.4, respectively. Again, from Figs. 12b and 12c, for some values of moving mass velocity, the linear analysis results are not in the safe side in predicting the maximum bending moment of the beam. Therefore, the nonlinear beam theory should be properly implemented in such cases. Fig. 8 Variation of the normalized contact force applied on an SS beam as a function of the moving mass weight for various values of the moving mass velocity. a Normalized minimum contact force vs. normalized moving mass weight; b Normalized maximum contact force vs. normalized moving mass weight Fig. 9 Variation of the normalized contact force applied on a CC beam as a function of the moving mass weight for various values of the moving mass velocity. a Normalized minimum contact force vs. normalized moving mass weight; b Normalized maximum contact force vs. normalized moving mass weight

10 On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method 173 Fig. 1 Variation of the normalized contact force applied on an SC beam as a function of the moving mass weight for various values of the moving mass velocity. a Normalized minimum contact force vs. normalized moving mass weight; b Normalized maximum contact force vs. normalized moving mass weight Fig. 11 Variation of the normalized contact force applied on a CF beam as a function of the moving mass weight for various values of the moving mass velocity. a Normalized minimum contact force vs. normalized moving mass weight; b Normalized maximum contact force vs. normalized moving mass weight Determination of the accuracy levels of the linear beam theory as the beam undergoes large deflections would be of great importance in the dynamic analyses of beam-like structures for practical applications. In this regard, the relative error parameter is defined as: e rel X linear X nonlinear / X linear, where the parameter X is the value of the appropriate design parameter. In the plane of M N V N, the contour plots for e rel.4,.6,.8, and.1 are given in Figs for various boundary conditions. It should be noted that the region placed between two arbitrary contour lines signifies

11 174 K. Kiani, A. Nikkhoo the normalized values of the mass weight and velocity of the moving mass in which the linear beam theory could predict the results of the nonlinear beam theory in the range of the specified accuracy levels. Moreover, the high accuracy level of the linear beam theory (e.g., e rel <.4) develops as one moves from CF to SS, then SC, and finally to CC boundary condition. However, the low accuracy level (e.g., e rel >.1) shrinks as one moves from CF to CC, then SC, and finally to SS boundary condition. In other words, the denser the contour lines are, the more sensitive the accuracy level to the variation of the moving mass weight and velocity are. Fig. 12 Effect of the moving mass velocity on the maximum deflection and bending moment of an SS beam for various values of the moving mass weight. a M N.1; b M N.2; c M N.4 Fig. 13 Effect of the moving mass velocity on the maximum deflection and bending moment of a CC beam for various values of the moving mass weight. a M N.1; b M N.2; c M N.4

12 On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method 175 Fig. 14 Effect of the moving mass velocity on the maximum deflection and bending moment of an SC beam for various values of the moving mass weight. a M N.1; b M N.2; c M N.4 Fig. 15 Effect of the moving mass velocity on the maximum deflection and bending moment of a CF beam for various values of the moving mass weight. a M N.1; b M N.2; c M N.4 5 Conclusions The discrete equations of motion for an elastically supported thin beam under the excitation of a moving mass were developed based on the assumptions of large deflections-small rotations of the beam structures. In this regard, an efficient meshless method, namely RKPM, was employed for spatial discretization of the problem and the Lagrange equations were applied to obtain the governing discrete equations of motion. The Crank Nicholson scheme was then adopted for time domain analysis of the resulting ordinary differential equations. A reasonably good agreement between the results of RKPM and those of Galerkin method was obtained for a simply supported boundary condition. The maximum values of deflection and bending moment were considered as the

13 176 K. Kiani, A. Nikkhoo Fig. 16 Contour plots of e rel for the normalized design parameters of an SS beam. a Normalized maximum displacement; b Normalized maximum bending moment Fig. 17 Contour plots of e rel for the normalized design parameters of a CC beam. a Normalized maximum displacement; b Normalized maximum bending moment crucial design parameters. The variations of design parameters associated with the linear and nonlinear beam theories were plotted in terms of moving mass weight and velocity for various boundary conditions. The results indicated that the values of design parameters of nonlinear beam theory are lower than those of linear beam theory in most of the studied cases. Furthermore, for the majority of the moving mass velocities, the discrepancies between the linear and nonlinear result become remarkable as the moving mass weight increases, specifically for high levels of moving mass velocity. It was observed that the distinction between the results of linear and nonlinear beam theories appears at certain values of moving mass velocity which varies with the beam boundary condition. Moreover, the possibility of mass separation from the base beam during excitation was studied. Except for the cantilever beam, the nonlinear beam theory predicts a higher

14 On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method 177 possibility of mass separation with respect to the linear one. Besides, in the nonlinear analysis, the magnitude of the minimum and maximum contact forces severely increases for higher levels of the moving mass weight and velocity. The accuracy levels of the linear beam theory for geometrically nonlinear beams were specified for various boundary conditions. It was shown that for a certain value of accuracy, the application limit of the linear beam theory mainly depends on the existing boundary conditions of the beam as well as on the moving mass weight and velocity. Fig. 18 Contour plots of e rel for the normalized design parameters of an SC beam. a Normalized maximum displacement; b Normalized maximum bending moment Fig. 19 Contour plots of e rel for the normalized design parameters of a CF beam. a Normalized maximum displacement; b Normalized maximum bending moment

15 178 K. Kiani, A. Nikkhoo Appendix (1) Computing the derivatives of L with respect to the u I, w I, u I, and ẇ I L E b A b φ u I,x u I α u uφ u I dγ, Γ b L lb w I [ u,x + 1 ] 2 (u2,x + w 2,x) (1 + u,x )dx [ {E b A b u,x + 1 ] 2 (u2,x + w 2,x) +E b I b w,xx φ w I,xx δ(x x M )H(l b x M )dx, L lb ρ b A b uφ u I u dx, I L lb ρ b (A b ẇφ w I + I b ẇ,x φ w I,x ẇ )dx. I (w,x φ w I,x ) } dx + Mφ w I [g (ẅ + 2vẇ,x + v 2 w,xx )] (2) Computing the elements of the matrix ˆf / z (A1) (A2) (A3) (A4) The partial derivatives of the ˆf with respect to the z could be obtained as f int f int ˆf z y x, (A5) I in which f int f int u, u f int u,ẇ y f int w, u f, f int int x f int u,u f int u,w w,ẇ f int w,u f, int w,w where the appropriate non-zero submatrices are defined as u,u] IJ u,w] IJ (A6) E b A b (1 + 3u,x +.5w 2,x)φ u I,xφ u J,xdx + α u φ u I φu JdΓ, (33) Γ b w,u] IJ w,w] IJ E b A b w,x (1 + u,x )φ u I,x φw J,x dx, E b A b w,x (1 + u,x )φ w I,x φu J,xdx, [E b A b (u,x +.5u 2,x + 1.5w 2,x) φ w I,x φw J,x + E bi b φ w I,xx φw J,xx ]dx + (K z φ w I φw J + K yφ w I,x φw J,x )dγ Γ b +Mv 2 φ w I (x M)φ w J,xx (x M)H(l b x M ), w,ẇ ] IJ 2Mvφ w I (x M)φ w J,x (x M)H(l b x M ). (3) Galerkin method formulation (A8) (A9) (A1) (A11) In this method, the unknown fields of the problem are discretized in the spatial domain of the beam as u(x, t) NM i1 a u i (t)ψ u i (x), w(x, t) NM i1 a w i (t)ψw i (x), (A12) in which a u i and a w i denote the unknown parameters that should be determined from the discrete equations of motion. The functions ψ u i and ψ w i are the i-th mode shapes associated with the longitudinal and transverse deformations which satisfy the boundary conditions of the problem for an elastic thin beam under free vibration, and NM is the number of modes. By substituting Eq. (A12) into Eqs. (16) (2), we get M uu M ww ρ b A b Υ uu, (A13) ρ b (A b Υ ww + I b Γ ww ) + Mψ w i (x M)ψ w j (x M)H(l b x M ), (A14) { f ext w } i Mgψ w i (x M)H(l b x M ), (A15) { f int u } i E b A b ( a u j Γuu ji au k au l au j Γuuuu kl ji au k au l Γuuu kli aw k aw l Γwwu kli + a u k au j Γuuu kji aw ma w n a u k Γwwuu mnki ( { fw int } i E b A b a u k aw j Γwwu k au k au l aw j Γwwuu kl +E b I b a w j Σww + M[v 2 ψ w i (x M)ψ w j,xx (x M)a w j +2vψ w i (x M)ψ w j,x (x M)ȧ w j ]H(l b x M ), where Υ uu Υ ww Σ ww Γ uu Γ ww Γ uuu k Γ wwu k Γ uuuu kl Γ wwuu kl ψ u i ψu j dx, ψ w i ψw j dx, ψ w i,xx ψw j,xx dx, ψ u i,xψ u j,xdx, ψ w i,x ψw j,x dx, ψ u i,x ψu j,x ψu k,x dx, ψ w i,x ψw j,x ψu k,x dx, ψ u i,x ψu j,x ψu k,x ψu l,x dx, ψ w i,x ψw j,x ψu k,x ψu l,x dx, ), (A16) aw k aw l aw j Γwwww kl ) (A17) (A18) in which 1 i, j, k, l NM. Therefore, the discrete equations of motion (i.e., Eq. (15)) are re-established according to RRM. It should be noted that the solving procedure in this method is similar to that of RKPM as explained in Sect. 3. In this regard, the submatrices of the matrix ˆf / z in RRM are obtained from Eqs. (A7) (A11) as follows u,u] E b A b ( Γ uu + 3a u k Γuuu kji aw k aw l Γwwuu kli j ), (A19) u,w] E b A b (a w k Γwwu kji + a w k au l Γwwuu kli j ), (A2) w,u] E b A b (a w k Γwwu ki j + a w k au l Γwwuu kil j ), (A21) ( w,w] E b A b (A22) a u k Γwwu k + a u k au l Γwwuu kl w,ẇ ] 2Mvψ w i (x M)ψ w j,x (x M)H(l b x M ) aw k aw l Γwwww kli j ) + E b I b Σ ww, (A23)

16 On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method 179 For a beam with simply supported-pinned movable boundary conditions, w(, t) w(l b, t), M b (, t) M b (l b, t), (A24) u(, t), N b (l b, t), it can be readily shown that the following mode shapes satisfy the conditions in Eq. (A24) [ (2i 1)πx ] ψ u i (x) sin, 2l b ( iπx ) ψ w i (x) sin, l b in this case, Eq. (44) is reduced to Υ uu Υ ww Σ ww Γ uu Γ ww Γ uuu k l b 2π { sin[(i j)π] i j sin[(i + j 1)π] i + j 1 }, l b[ j sin(iπ)cos(jπ) i cos(iπ)sin(jπ)], π(i 2 j 2 ) π[ j sin(iπ)cos(jπ) i cos(iπ)sin(jπ)], l b (i 2 j 2 ) π(2i 1)(2 j 1) { sin[(i + j 1)π] + 8l b i + j 1 Γ wwu k π[isin(iπ)cos(jπ) j cos(iπ)sin(jπ)], l b (i 2 j 2 ) { cos[(i j k)π] 2i 2 j 2k + 1 (2i 1)(2 j 1)(2k 1)π2 16l 2 b cos[(i + j k)π] cos[(i j k)π] 2i + 2 j 2k 1 2i 2 j 2k 1 sin[(i + j + k 1.5)π] + 2i + 2 j + 2k 3 (2k 1)π2 4l 2 b cos[(i j + k)π] 2i 2 j + 2k + 1 }, sin[(i j)π] }, i j { cos[(i j k)π] cos[(i + j k)π] + 2i 2 j 2k + 1 2i + 2 j 2k + 1 i j + k + l 1 sin[(i + j + k.5)π] 2i + 2 j + 2k 1 (A25) }, (A25) Γ uuuu (2i 1)(2 j 1)(2k 1)(2l 1)π3 kl 18l 3 b { sin[(i j k l + 1)π] sin[(i + j k l)π] + i j k l + 1 i + j k l sin[(i j + k l)π] sin[(i + j + k l 1)π] + + i j + k l i + j + k l 1 sin[(i j k + l)π] sin[(i + j k + l 1)π] + + i j k + l i + j k + l 1 sin[(i j + k + l 1)π] sin[(i + j + k + l 2)π] } + +, Γ wwuu kl i + j + k + l 2 { sin[(i j k l + 1)π] i j k l + 1 (2k 1)(2l 1)π3 32l 3 b sin[(i + j k l + 1)π] sin[(i j + k l)π] + + i + j k l + 1 i j + k l sin[(i + j + k l)π] sin[(i j k + l)π] + + i + j + k l i j k + l sin[(i + j k + l)π] sin[(i j + k + l 1)π] + + i + j k + l i j + k + l 1 + sin[(i + j + k + l 1)π] i + j + k + l 1 }, for the formulations in Eq. (A25), in the case of indeterminate limit form of type /, the L Hopital s rule is employed. References 1 Frýba, L.: Vibration of Solids and Structures under Moving Loads. Thomas Telford, London (1999) 2 Hino, J., Yoshimura, T., Ananthanarayana, N.: Vibration analysis of non-linear beams subjected to a moving load using the finite element method. J. Sound Vib. 1(4), (1985) 3 Yoshimura, T., Hino, J., Anantharayana, N.: Vibration analysis of a non-linear beam subjected to moving loads by using the galerkin method. J. Sound Vib. 14(2), (1986) 4 Wang, R.T., Chou, T.H.: Non-linear vibration of Timoshenko beam due to a moving force and the weight of beam. J. Sound Vib. 218(1), (1998) 5 Lee, U.: Revisiting the moving mass problem: onset of separation between the mass and beam. J. Vib. Acoust. ASME 118(3), (1996) 6 Stancioiu, D., Ouyang, H., Mottershead, J.E.: Vibration of a beam excited by a moving oscillator considering separation and reattachment. J. Sound Vib. 31, (29) 7 Nikkhoo, A., Rofooei, F.R., Shadnam, M.R.: Dynamic behavior and modal control of beams under moving mass. J. Sound Vib. 36(3-5), (27) 8 Kiani, K., Nikkhoo, A., Mehri, B.: Prediction capabilities of classical and shear deformable beam models excited by a moving mass. J. Sound Vib. 32(3), (29) 9 Kiani, K., Nikkhoo, A., Mehri, B.: Parametric analyses of multispan viscoelastic shear deformable beams under excitation of a moving mass. J. Vib. Acoust. ASME 131(5), 519(1-12) (29) 1 Kiani, K., Nikkhoo, A., Mehri, B.: Assessing dynamic response of multispan viscoelastic thin beams under a moving mass via generalized moving least square method. Acta Mech. Sin. 26(5) (21) 11 Xu, X., Xu, W., Genin, J.: A non-linear moving mass problem. J. Sound Vib. 24(3), (1997) 12 Siddiqui, S.A.Q., Golnaraghi, M.F., Heppler, R.: Dynamics of a flexible cantilever beam carrying a moving mass. Nonlinear Dyn. 15, (1998) 13 Liu, W.K., Jun, S., Zhang, Y.F.: Reproducing kernel particle methods. Int. J. Numer. Meth. Fluids 38, (1995) 14 Liu, W.K., Chen, Y., Jun, S., et al.: Overview and applications of the reproducing kernel particle methods. Arch. Comp. Meth. Eng. 3, 3 8 (1996) 15 Liu, W.K., Chen, Y., Chang, C.T., et al.: Advances in multiple scale kernel particle methods. Comp. Mech. 18(2), (1996) 16 Liu, W.K., Jun, S., Li, S., et al.: Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Meth. Eng (1995) 17 Kiani, K.: A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect. Int. J. Mech. Sci (21) 18 Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. Volume 2: Solid Mechanics. Fifth edition, Butterworth- Heinemann Publisher, Oxford (2)