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A consistent co-rotational formulation and numerical procedure for the dynamic analysis of planar Timoshenko beam is presented in detail to demonstrate the finite element analysis techniques of co-rotational formulation for nonlinear dynamic analysis of beam structures. The inertia nodal forces and deformation nodal forces are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory by using the d’Alembert principle and the virtual work principle. The nodal coordinates, incremental displacements and rotations, velocities, accelerations and the equations of motion of the system are defined in terms of a fixed global coordinate system, while the total strains in the beam element are measured in element coordinates which are constructed at the current configuration of the beam element. A practical numerical procedure is proposed to determine the current end cross-section coordinates, element coordinates, and element deformations corresponding to an incremental displacement. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the 284nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the effectiveness of the proposed method and investigate the effect of the shear deformafion on the dynamic response of the beam structures.

An explicit method for nonlinear transient dynamic analysis of spatial beams with finite rotations using a corotational total Lagrangian finite element formulation is presented. The kinematics of the beam element is described in the current element coordinate system constructed in the current configuration of the beam element. The element deformation and inertia nodal forces are derived by the virtual work principle, the d'Alembert principle, and the consistent linearization of the geometrically nonlinear beam theory. A nodal rotation vector is used to represent the finite rotation of a base coordinate system rigidly attached to each node of the discretized structure. A numerical procedure of explicit method is proposed for the solution of the nonlinear equations of motion. The standard central difference method is applied to the incremental displacement vector and the incremental rotation vector, and the time derivatives of displacement vector and rotation vector. The nodal orientations are updated by the incremental nodal rotation vectors. The values of nodal rotation vectors are reset to zero in the current configuration. In order to assess the efficiency and the accuracy of the proposed method, numerical examples are studied and compared with the results obtained using the implicit method based on the Newmark method.

The axially loaded thin-walled beams with point symmetric open section are studied using a corotational finite element formulation. The kinematics of the beam element is described in the current element coordinate system. The element nodal forces are derived using the virtual work principle, and consistent second order linearization of the fully geometrically non-linear beam theory. Different axial load systems with the same centric resultant axial force but different resultant bimoments are considered. Numerical examples studied show that the effects of bimoments on the buckling and postbuckling behavior of point symmetric open section beams are remarked.

A 4 degree-of-freedom (DoF) benchmark model of multiple bifurcation (MB) in compound stability problems of nonlinear structures is proposed. In the MB model, the governing equations are designed to be highly simplified when two critical modes exist in the singular Jacobian matrix. The resulting MB equations can be solved analytically through manual calculation. To verify the applicability, graphical solution methods are applied to solve the MB equations and visualize the multiple path branching through a graphical monitor. The proposed benchmark model and graphical strategies can help understand the stability phenomenology and MB in the stability design of large-scale finite element models.

The optimization of one-phase dimensional synthesis problems for path-generating four-bar mechanisms is a difficult task. Such problems have mainly been solved in the literature using the differential evolution (DE) algorithm and its variants, such as a combined-mutation DE (CMDE). The shortcoming of using DE or its variants lies in the time-consuming parameter tuning that can take up to a dozen iterations. In this study, two well-known population-based metaheuristic optimization methods, i.e. the cuckoo search (CS) and teaching–learning-based optimization (TLBO) algorithms are employed to solve five representative problems. Only one user-supplied parameter is needed for CS and none for TLBO. Findings show that the solution accuracy of CS is superior to that of DE and TLBO. Moreover, CS, with its advantage of parameter tuning only four times, rivals CMDE in solution accuracy. The reason for the superior exploitation capability of CS lies in the special combination of two search modes, i.e. the Lev...

A curved slotted Geneva mechanism can eliminate the adversely infinite angular jerks of the Geneva wheel and might reduce the peak angular acceleration of the Geneva wheel by using a proper indexing motion program. In the literature, the cycloidal, fifth-order polynomial and modified sine indexing motion programs are frequently used for curved slotted Geneva mechanisms. To achieve the better kinematic performance of the curved slotted Geneva wheel than that obtained using the above-mentioned indexing motion programs, a new indexing motion program based on the Hermite interpolating polynomial is proposed for an optimum design with the goals of minimizing the peak angular acceleration and eliminating the adversely infinite angular jerks. The domain of the indexing position function is divided into several segments. Each segment is termed an element, and both ends of each segment are termed nodes. The nodal values of the indexing position function and its derivatives are used as design variables. The position function for each element can be described using the Hermite interpolating polynomial and the design variables. The reason behind the use of the Hermite interpolating polynomial is that the design variables have the clear physical meanings. The four-level Hermite interpolating polynomial is used and two elements are sufficient to obtain the optimum results. In addition, the constraint regarding the radius of curvature of the profile of the inner slot is proposed to prevent sharp curvature of the profile of the inner slot. The findings show that there is a decline in the peak acceleration of the Geneva wheel with six curved slots for the optimum results obtained using the proposed indexing motion program by 33.4% and 24.3%, respectively, as compared with the cycloidal and modified sine indexing motion programs.

When the traditional grasping index is used as an objective function, the optimized mechanism obtained offers good force transmission performance but unsound motion transmission for a 10-link gripping mechanism. Therefore, we first propose a new grasping index that is the product of the traditional grasping index and the effective force ratio. The effective force ratio reflects the motion transmission for the output links. The proposed grasping index is similar to the force transmission index discussed in the literature; however, different views of the physical meaning of the two components of the index and the necessity of the effective force ratio for optimum synthesis of the mechanism are presented. The second new grasping index proposed is the product of the first grasping index and the cosine value of the angle between the contact force and the contact velocity, originally used as a constraint. Using these new grasping indices, one can obtain sound optimum mechanism designs with a very small ...

An optimum design of variable input speed for the Geneva mechanism is aimed at improving the kinematic performance of the traditional Geneva mechanism by eliminating infinite angular jerks and reducing the peak angular acceleration of the Geneva wheel during the indexing motion. The normalized angular velocity and acceleration of the Geneva wheel corresponding to the normalized time are introduced. A polynomial function of the normalized time is used to describe the normalized angular position of the crank, and therefore, the corresponding polynomial coefficients are considered as the design variables. The optimum design task is very specialized and difficult to solve with some evolutionary and swarm optimization methods because of the extremely large range for the value of the design variable, arising from the utilization of a higher order polynomial for the normalized time parameter with a value between 0 and 1. A new evolutionary algorithm termed teaching-learning-based optimization comprises a teacher phase and a learner phase. In the teacher phase, the entire population can be gradually shifted to a more promising region, which may be very far from the relatively small initial region. The obtained optimal results are compared with those obtained using the length-adjustable deriving link method discussed in the literature. The findings show that the difference in the effectiveness of the variable input speed method and the length-adjustable driving link method for the reduction of the peak angular acceleration of the Geneva wheel is small.

A new micro single flute drill was presented in this study. For the convenience of developing a solid model, and doing a geometrical analysis, its geometry including the flute and flank surfaces is presented mathematically. According to the geometrical analysis, the normal clearance angles and normal rake angles along the cutting lip of the new micro dill are always invariant, and positive, respectively. Hence its cutting force in drilling is reduced. Furthermore, with only one main flute, the new micro drill gets stronger, and its cutting torque can be also decreased. Therefore it is expected that the tool life of the new micro drill is much longer than that of the conventional drill. Keyword: New micro drill, micro drill, single cutting edge

The objective of this study is to investigate the deformed configuration and free vibration around the deformed configuration of clamped buckled beams by co-rotational finite element formulation. The principle of virtual work, d'Alembert principle and the consistent second order linearization of the nonlinear beam theory are used to derive the element equations in current element coordinates. The governing equations for linear vibration are obtained by the first order Taylor series expansion of the equation of motion at the static equilibrium position of the buckled beam. Numerical examples are studied to investigate the natural frequencies of clamped buckled beams with different slenderness ratios under different axial compression.

The steady state deformation and infinitesimal free vibration around the steady state deformation of a rotating inclined Euler beam at constant angular velocity are investigated by the corotational finite element method combined with floating frame method. The element nodal forces are derived using the consistent second order linearization of the nonlinear beam theory, the d'Alembert principle and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The rotating element coordinates rotate about the hub axis at the angular speed of the hub. The equations of motion of the system are defined in terms of an inertia global coordinate system, which is coincident with a rotating global coordinate system rigidly tied to the rotating hub. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady state deformation and natural frequency of the rotating inclined beam.

A corotational finite element method combined with floating frame method and a numerical procedure is proposed to investigate large steady-state deformation and infinitesimal-free vibrationaround the steady-state deformation of a rotating-inclined Euler beam at constant angular velocity. The element nodal forces are derived using the consistent second-order linearization of the nonlinear beam theory, the d'Alembert principle, and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The governing equations for linear vibration are obtained by the first-order Taylor series expansion of the equation of motion at the position of steady-state deformation. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady-state deformation and natural frequency of the rotating beam with different inclined angle, angular velocities, radius of the hub, and slenderness ratios.

The natural frequency of the flapwise bending vibration, and coupled lagwise bending and axial vibration is investigated for the rotating beam. A method based on the power series solution is proposed to solve the natural frequency of very slender rotating beam at high angular velocity. The rotating beam is subdivided into several equal segments. The governing equations of each segment are solved by a power series. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method. The effect of Coriolis force, angular velocity, and slenderness ratio on the natural frequency of rotating beams is investigated.

The coupled vibration of thin-walled beams with a generic open section induced by the boundary conditions is investigated using the finite element method. If the axial displacement of the pin end is restrained at another point rather than the centroid of the asymmetric cross section, the axial vibration, two bending vibrations, and torsional vibration may be all coupled. The element developed here has two nodes with seven degrees of freedom per node. The shear center axis is chosen to be the reference axis and the element nodes are chosen to be located at the shear centers of the end cross sections of the beam element. Different sets of element nodal degrees of freedom corresponding to different pin ends are considered here. The relation between element matrices referred to different sets of element nodal degrees of freedom is derived. Numerical examples are presented to demonstrate the accuracy of the proposed method and to investigate the effects of different pin ends on the coupled vibrations of the thin-wall beam.

The coupled axial–torsional vibration of thin-walled Z-section beam induced by the boundary conditions is investigated. The value of the warping function is not zero at centroid for Z-section beam. If the axial displacement of the pin end is restrained at the centroid of the Z-section for thin-walled Z-section beam, the axial vibration and torsional vibration may be coupled. The governing equations for linear axial and torsional vibration of a thin-walled Z-section beam are derived by the d’Alembert principle and the virtual work principle. The bending vibration is uncoupled from axial and torsional vibrations and is not dealt with in this paper. For harmonic vibration, the general solution of these equations with undetermined constant coefficients may be obtained. Substituting the general solution into the displacement and force boundary conditions, a set of homogeneous equations can be obtained. The natural frequencies and the coefficients of the general solution may be obtained by solving the homogeneous equations using the bisection method. Numerical examples are studied to verify the accuracy of the proposed method and to investigate the effect of boundary conditions and the value of warping function at centroid on the coupled axial and torsional natural frequency of Z-section beam.

A consistent co-rotational total Lagrangian finite element formulation for the geometric nonlinear buckling and postbuckling analysis of thin-walled beams with generic open section is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross-sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system constructed at the current configuration of the beam element. The element nodal forces are derived using the virtual work principle. The virtual rigid body motion corresponding to the virtual nodal displacements is excluded in the derivation of the element nodal forces. A procedure is proposed to determine the virtual rigid body motion. The way used to determine the element coordinate system and element nodal deformations corresponding to the virtual nodal displacements and that corresponding to the incremental nodal displacement are consistent. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. In the derivation of the element tangent stiffness matrix, the change of element nodal forces induced by the element rigid body rotations should be considered for the present method. Thus, a stability matrix is included in the element tangent stiffness matrix. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. Numerical examples are presented to investigate the accuracy and efficiency of the proposed method. The effect of the terms in the element nodal force and tangent stiffness matrix, which will converge to zero with the decrease of element size, on the convergence rate of solution and accuracy for the buckling load and nonlinear behavior of three dimensional beam structures are also investigated through numerical examples. � 2005 Elsevier B.V. All rights reserved.

In this study, an analytical formulation for the necessary thrust of the crosshead of the five-point double-toggle clamping mechanism during real-mold clamping operation is proposed in order to investigate the effect of the friction at pin joints. The friction is considered to be a Coulomb friction. The inertia effects are not considered. Numerical examples are studied for different values of coefficient of friction to investigate the effects of friction at pin joints. In order to estimate the kinetic friction coefficient at pin joints, the motor torques required for real-mold clamping obtained by the present study are adjusted by using different values of coefficient of friction and mechanical efficiency to fit the experimental data.

In this study, an analytical formulation for the torsional warping function of a thin-walled open-section beam is built based on the combination of the Vlasov assumption and the Kirchhoff assumption of plate/shell theory, essentially due to Goodier and Gjelsvik. Vlasov, Timoshenko and many authors (follow Vlasov) only consider the contour warping function as the real warping function. Goodier and Gjelsvik (follow Goodier) not only consider the contour warping but also the thickness warping. For some cross-sectional constants related to the warping function such as the torsional constant, the adoption of the warping function based on Vlasov's theory or Timoshenko's theory may cause them incorrectness. On the other hand, the torsional constant based on the Goodier's theory is consistent with the one based on the membrane analogy. Thus, Goodier's theory is a good approximation for the torsional warping of a thin-walled open-section beam. To the authors’ limited knowledge, the analytical expression for the complete torsional warping of a thin-walled open-section beam has not been found in the literature. In this study, a more general expression for the torsional warping of a thin-walled open-section beam is presented. The position formulas to determine the twist center are also given.

A doubly symmetric thin-walled beam element with open section is derived using co-rotational (CR) total Lagrangian (TL) formulation. The effects of deformation-dependent third-order terms of element nodal forces on the buckling load and post-buckling behavior are investigated. All coupling among bending, twisting, and stretching deformations for beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, all third-order terms of nodal forces, which are relevant to the twist rate, rate of twist rate and curvature of the beam axis are also considered. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A parabolic interpolation method of the arc length is used to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed element and to investigate the effect of third-order terms of element nodal forces on the buckling load and post-buckling behavior of doubly symmetric thin-walled beams.

A consistent co-rotational total Lagrangian finite element formulation and numerical procedure for the geometric nonlinear buckling and postbuckling analysis of thin-walled beams with monosymmetric open section is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross-sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system, which is constructed at the current configuration of the beam element. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, the third-order term of the twist rate of the beam axis is considered in element nodal forces. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A parabolic interpolation method of the arc length is used to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A consistent co-rotational finite element formulation and numerical procedure for the buckling and postbuckling analyses of three-dimensional elastic Euler beam is presented. All coupling among bending, twisting, and stretching deformations for a beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, the third-order terms, which are relevant to the twist rate and curvature of the beam axis, are also considered. An incremental–iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A bisection method of the arc length is proposed to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method and to investigate the effect of third-order terms on the buckling load and postbuckling behavior of three-dimensional beams.

A co-rotational total Lagrangian finite element formulation for the geometrically nonlinear dynamic analysis of spatial Euler beam with large rotations but small strain, is presented. The nodal coordinates, displacements, rotations, velocities, accelerations, and the equations of motion of the structure are defined in a fixed global set of coordinates. The beam element has two nodes with six degrees of freedom per node. The element nodal forces are conventional forces and moments. The kinematics of beam element are defined in terms of element coordinates, which are constructed at the current configuration of the beam element. Both the element deformation nodal forces and inertia nodal forces are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory using the d'Alembert principle and the virtual work principle in the current element coordinates. An incremental-iterative method based on the Newmark direct integration method and the Newton—Raphson method is employed here for the solution of the nonlinear equations of motion. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A consistent co-rotational finite element formulation and numerical procedure for the linear buckling analysis of three-dimensional elastic Euler beam is presented. A mechanism for generating configuration dependent conservative moment is proposed and the corresponding load stiffness matrix is derived. It is assumed that the prebuckling displacements and rotations of the structure and the corresponding deformations of the elements are linearly proportional to the external loading. The prebuckling rotations of the structure are fixed axis rotations or small rotations, and the effect of the prebuckling displacement on transformation matrix for the coordinates transformation can be ignored. All coupling among bending, twisting and stretching deformations for beam element is considered by consistent linearization of the fully geometrically nonlinear beam theory. An inverse power method for the solution of the generalized eigenvalue problem is used to find the buckling load and buckling mode. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A co-rotational finite element formulation for the dynamic analysis of a planar curved Euler beam is presented. The Euler-Bernoulli hypothesis and the initial curvature are properly considered for the kinematics of a curved beam. Both the deformational nodal forces and the inertial nodal forces of the beam element are systematically derived by consistent linearization of the fully geometrically nonlinear beam theory in element coordinates which are constructed at the current configuration of the corresponding beam element. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear dynamic equilibrium equations. Numerical examples are presented to demonstrate the effectiveness of the proposed element and to investigate the effect of the initial curvature on the dynamic response of the curved beam structures.

The geometrically nonlinear analysis of laminated composite rectangular plates is studied using the hierarchical finite element method (HFEM). The derivation of the equilibrium equations and tangential stiffness matrix are given. Symbolic computation is used to calculate the high-order polynomial integrals needed to establish the stiffness matrices. The Newton-Raphson method is used in the iterative procedure. The convergence property and the numerical stability of the method are discussed. The influence of in-plane displacements on the geometrically nonlinear deformation is also discussed. The results of static analyses indicate that the extension of HFEM to geometrically nonlinear analysis of laminated rectangular plates is very successful. It is believed that this scheme can be easily applied to geometrically nonlinear dynamic analysis of laminated plates.

This paper deals with the transient vibration of asymmetric rotor systems during acceleration passing through several critical speeds at which synchronous or super-harmonic resonance occurs. The dynamic equations of the rotor-bearing system are formulated by the finite element model and the resulting dynamic equations are time-varying due to the effects of acceleration and asymmetry. In the formulation, a Timoshenko beam element is employed to simulate the rotating shaft and Eulerian angles are used to describe the orientations of the shaft element and disk. The numerical integration scheme for transient analysis is generated from the finite element model. Numerical examples are presented to illustrate (1) the effects of acceleration on peak amplitude and speed at which the peak occurs as the system passes through critical speeds, (2) the optimal acceleration process, which can be obtained by minimizing the peak response and the period of acceleration, (3) the speed regions where the transient instability exists.

A method based on the power series solution is proposed to solve the natural frequency of flapping vibration for the rotating inclined Euler beam with constant angular velocity. The vibration of the rotating beam is measured from the position of the corresponding steady state axial deformation. In this paper the governing equations for linear vibration of a rotating Euler beam are derived by the d'Alembert principle, the virtual work principle and the consistent linearization of the fully geometrically nonlinear beam theory in a rotating coordinate system. The governing equation for flapping vibration of the rotating inclined Euler beam is linear ordinary differential equation with variable coefficients and is solved by a power series with four independent coefficients. Substituting the power series solution into the corresponding boundary conditions at two end nodes of the rotating beam, a set of homogeneous equations can be obtained. The natural frequencies may be determined by solving the homogeneous equations using the bisection method. Numerical examples are studied to investigate the effect of inclination angle on the natural frequency of flapping vibration for rotating inclined Euler beams with different angular velocity and slenderness ratio., {"references":["M. J. Schilhansl Bending frequency of a rotating cantilever beamASME Journal of Applied Mechanics vol. 25 pp. 28-30, 1958.","J. T. S. Wang, O. Mahrenholtz and J. Bohm, Extended Galerkin's method for rotating beam vibrations using Legendre polynomials SolidMechanics Archives, vol. 1, pp. 341-365, 1976.","A. Leissa Vibrational aspects of rotating turbomachinery bladesASME Applied Mechanics Reviews vol. 34 pp. 629-635, 1981.","D. H. Hodges and M. J. Rutkowski, Free-vibration analysis of rotating beams by a variable-order finite-element method, AIAA Journal, vol. 19, pp. 1459-1466, 1981.","T. Yokoyama Free vibration characteristics of rotating Timoshenko beam International Journal of Mechanical Science vol. 30 pp. 743-755, 1988.","H. H. Yoo and S. H. Shin, Vibration analysis of rotating cantilever beams Journal of Sound and vibration, vol. 212, pp. 807-828, 1988.","S. Y. Lee and Y. H. Kuo Bending frequency of a rotating beam with an elastically restrained root ASME Journal of Applied Mechanics vol. 58pp. 209-214, 1991.","H. P. Lee, Vibration on an inclined rotating cantilever beam with tip mass, ASME Journal of Vibration and Acoustics, vol. 115, pp. 241245,1993.","S. C. Lin and K. M. Hsiao Vibration analysis of rotating Timoshenko beam Journal of Sound and Vibration vol. 240, pp. 303-322, 2001. [10] A. A. Al-Qaisia, Non-linear dynamics of a rotating beam clamped with an attachment angle and carrying an inertia element, The Arabian Journal for Science and Engineering, vol. 29, pp. 81-98, 2004. [11] S. Y. Lee and J. J. Sheu, Free vibrations of a rotating inclined beam, ASME Journal of Applied Mechanics vol. 74 pp. 406-414, 2007. [12] S. Y. Lee and J. J. Sheu, Free vibration of an extensible rotating inclined Timoshenko beam, Journal of Sound and Vibration, vol. 304, pp. 606-624, 2007. [13] J. C. Simo and K. Vu-Quac The role of non-linear theories in transient dynamic analysis of flexible structures Journal of Sound and Vibrationvol. 119, pp. 487-508, 1987. [14] K. M. Hsiao, Corotational total Lagrangian formulation for three-dimensional beam element, AIAA Journal, vol. 30, pp. 797-804, 1992.[15] K. M. Hsiao, R. T. Yang, and A. C. Lee, A cosistent finite element formulation for nonlinear dynamic analysis of planar beam, International Journal for Numerical Methods in Engineering, vol. 37, pp. 75-89, 1994 [16] P. W. Likins, Mathematical modeling of spinning elastic bodies for model analysis AIAA Journal, vol. 11, pp. 1251-1258, 1973."]}

A motion process of a spatial beam element which can retain the continuity of spatial frames at rigid joints is poroposed to remove the restriction of small rotations between two successive increments for large displacement analysis of spatial frames using a corotational formulation combined with incremental-iterative methods. Two different strategies are proposed to determine the current configuration. The element formulation is derived based on the small deflection beam theory with the inclusion of the effect of axial force in the element coordinate system which is determined by the locations of end nodes and the orientations of end cross sections of the beam element. The element deformations are determined by rotations of end cross sections relative to the element coordinate system. An incremental-iterative method based on the Newton-Raphson method combined with are length control is adopted. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.

A co-rotational formulation of beam element and numerical procedure for the dynamic analysis of planar flexible mechanisms is presented. The nodal coordinates, velocities, accelerations, incremental displacements and rotations, and equations of motion of the system are defined in terms of fixed global coordinates, while the total strains in the beam element are measured in an element coordinate system which rotates and translates with the element but does not deform with the element. The element equations are constructed using the small deflection beam theory with the inclusion of the effect of axial force first in the element coordinate system, and then transformed to the global coordinate system using a standard procedure. In the proposed approach the resulting equations of motion are the same as those typically arising in nonlinear strctural dynamics. An incremental-iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for numerical studies. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.

The objective of this paper is to derive an exact expression for linear vibration equations of a rotating Euler beam, and investigate the effect of Coriolis and centrifugal force on the natural frequency of the rotating beam. Consider a uniform Euler beam rigidly mounted on the periphery of a rigid hub, which rotates about the hub axis fixed in space at a constant angular speed. The deformational displacements of the beam are defined in a rotating rectangular Cartesian coordinate system which is rigidly tied to the hub. The vibration of the beam is measured from the position of the steady‐state axial deformation, and only infinitesimal free vibration is considered. The equations of motion for the rotating Euler beam are derived by the d’Alembert principle and the virtual work principle. In order to capture all inertia effect and coupling between extensional and flexural deformation, the consistent linearization of the fully geometrically nonlinear beam theory is used in the derivation. A method based on the power series solution is proposed to solve the natural frequency. Numerical examples are studied for the natural frequency of rotating beams with different angular velocity and setting angle.

A tension\char22{}Frenkel-Kontorova dislocation model with a nonlinear interatomic force appears to be better justified than previous models in the interpretation of the Peierls stress of dislocation motion. Numerical anisotropic calculations on several fcc crystals result in good agreement within the order of Seeger's double-kink-generation interpretation of Bordoni relaxation peaks. Comparisons of the recalculation results of the Peierls stress with Hobart's compression model suggest that the present tension model is a more realistic representation of an edge-dislocation migration, in which energy is thought to be concentrated on these tension atoms just below the slip plane. It is also possible with this analysis to estimate the surface energies of material studied by internal friction in close agreement. This suggests that the tension\char22{}Frenkel-Kontorova model is more than a phenomenological representation of an edge dislocation.

A corotational total Lagrangian (CR-TL) formulation using the Von Karman assumption is presented for the analysis of nonlinear shell problems. This formulation is applicable for the nonlinear analysis of shell structures with large rotations but moderate deformations. The numerical algorithm used here is an incremental iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector. In order to improve the convergence properties of the equilibrium iterations, special care is taken in the calculation of the geometric stiffness matrix. Numerical examples are presented and compared with the results reported in the literature to demonstrate the accuracy and efficiency of the proposed method.

A simple and practical formulation of beam element for the large displacement analysis of elasto-plastic plane frames is presented. A corotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. A body attached coordinate is used to distinguish between rigid body and deformational rotations. The deformational nodal rotational angles are assumed to be small, and the membrane strain along the deformed beam axis is assumed to be constant. The element strain and stress are calculated using the total deformational nodal rotations in the body attached coordinate. The element internal nodal force vector is obtained from the virtual work principle. The element stiffness matrix is established from the derivative of the element internal nodal force vector with respect to the element nodal displacement vector. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length control method is employed for the solution of the nonlinear equilibrium equations. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A very simple and effective formulation and numerical procedure to remove the restriction of small rotations between two successive increments for the geometrically nonlinear finite element analysis of in-plane frames is presented. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. A body attached coordinate is used to distinguish between rigid body and deformational rotations. The deformational nodal rotational angles are assumed to be small, and the membrane strain along the deformed beam axis obtained from the elongation of the arc length of the deformed beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations in the body attached coordinate. The element stiffness matrix is obtained by superimposing the bending and the geometric stiffness matrices of the elementary beam element and the stiffness matrix of the linear bar element. An incremental iterative method based on the Newton-Raphson method combined with a constant arc length control method is employed for the solution of the nonlinear equilibrium equations. In order to improve convergence properties of the equilibrium iteration, a two-cycle iteration scheme is introduced. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.