Variable Kinematics Research Articles

Refined finite element solutions for anisotropic laminated plates

This paper presents some solutions for mechanical responses of angle-ply laminated plates under transverse distributed loads, which are obtained by using refined finite element models adopting variable kinematics based on Carrera’s Unified Formulation (CUF). Plates with several types of stacking sequence under different boundary conditions are considered. Layer-wise (LW) models based on Chebyshev polynomials (first kind) and Equivalent Single Layer (ESL) models based on Trigonometric series are used in the analysis. To compare the performances of different displacement-based kinematic models, a set of simply supported boundary conditions and mixed clamped-free boundaries are adopted in the numerical study. A nine-node MITC (Mixed Interpolated of Tensorial Components) plate element is employed to contrast the shear locking phenomenon of thin plates. CUF-based variable kinematic models are used in the numerical study and the number of expansion terms in the thickness direction is increased until the requisite numerical accuracy is achieved. By comparing the numerical results obtained with CUF-based refined models and ABAQUS 3D models as well as reference solutions from literature, the effectiveness of the adopted models is verified. The newly studied numerical cases can be taken as benchmarks for future research.

Specifičnosti procesa namotavanja trakastog materijala u obliku asure od sečenih staklenih vlakana (MAT-a)

The processes of winding (unwinding) are characterized by variable kinematic and dynamic parameters. Thus, in the case of increasing the diameter of the coiled material, its frequency of rotation is reduced, and vice versa. The implemented system is also specific in the manner of transmission of torque, using friction transmission. For these reasons, this paper mathematically defines the variable kinematic and dynamic drive parameters using the example of matting winding from staple glass fibers at ETEKS - from Baljevac. To make the process of winding to work through constant tractive force and linear speed, the power on the traction roller (winding) has to be constant. It must be such that it provides sliding between internalinterlayers as well as avoiding the severance (tearing) of the base material. The obtained dependencies are simulated on a computer, for matting of 450g/m2 grammage, and the results showed a significant agreement with the results of weight measurements of the wound-up mattings of the projected operating device.

Shell elements with through-the-thickness variable kinematics for the analysis of laminated composite and sandwich structures

Abstract Several efforts have been made in the last years to improve the efficiency and the effectiveness of structural models for the analysis of laminated shell structures. Among the others, many recent and past works in the literature have been aimed at formulating theories of structures that maximize the accuracy of analysis meanwhile reducing the computational costs. In this paper, this objective is pursued by implementing advanced shell theories with through-the-thickness variable kinematic capabilities. By employing the Carrera Unified Formulation (CUF), the proposed shell model is obtained by expressing the displacement field as an arbitrary and, eventually, hierarchical expansion of the primary unknowns along the thickness. Thus, Equivalent-Single-Layer (ESL), Layer-Wise (LW) models as well as variable kinematic models which combine ESL and LW approaches within the shell thickness can be obtained in a straightforward and unified manner. After the unified shell model is formulated, the governing equations and the related finite element arrays are obtained by employing the principle of virtual work. A nine node finite element is implemented to approximate the solution field, and the Mixed Interpolation of Tensorial Components (MITC) method is used to contrast the membrane and shear locking phenomena. Some numerical examples are discussed, including three- and ten-layered cross-ply shells under bi-sinusoidal load and simply-supported boundary conditions, a multilayered spherical panel subjected to bi-sinusoidal load and a sandwich cylinder undergoing bi-sinusoidal pressure. Moreover, various thickness and radius-to-thickness ratios are considered. Whenever possible, the results are compared with those from the literature and from exact elasticity solutions. The analysis of the results clearly shows the enhanced capabilities of the present variable-kinematic shell element, which allows the analyst to opportunely reduce the computational costs and enhance the accuracy of the model only in those regions of the thickness domain where an accurate evaluation of the stress/strain field is needed.

Development of a bio-inspired transformable robotic fin

Fish swim by oscillating their pectoral fins forwards and backwards in a cyclic motion such that their geometric parameters and aspect ratios change according to how fast or slow a fish wants to swim; these complex motions result in a complicated hydrodynamic response. This paper focuses on the dynamic change in the shape of a fin to improve the underwater propulsion of bio-inspired mechanism. To do this, a novel transformable robotic fin has been developed to investigate how this change in shape affects the hydrodynamic forces acting on the fin. This robotic fin has a multi-link frame and a flexible surface skin where changes in shape are activated by a purpose designed multi-link mechanism driven by a transformation motor. A drag platform has been designed to study the performance of this variable robotic fin. Numerous experiments were carried out to determine how various controlling modes affect the thrust capability of this fin. The kinematic parameters associated with this robotic fin include the oscillating frequency and amplitude, and the drag velocity. The fin has four modes to control the cyclic motion; these were also investigated in combination with the variable kinematic parameters. The results will help us understand the locomotion performance of this transformable robotic fin. Note that different controlling modes influence the propulsive performance of this robotic fin, which means its propulsive performance can be optimized in a changing environment by adapting its shape. This study facilitates the development of bio-inspired unmanned underwater vehicles with a very high swimming performance.

Design and accuracy analysis of a metamorphic CNC flame cutting machine for ship manufacturing

The current research of processing large size fabrication holes on complex spatial curved surface mainly focuses on the CNC flame cutting machines design for ship hull of ship manufacturing. However, the existing machines cannot meet the continuous cutting requirements with variable pass conditions through their fixed configuration, and cannot realize high-precision processing as the accuracy theory is not studied adequately. This paper deals with structure design and accuracy prediction technology of novel machine tools for solving the problem of continuous and high-precision cutting. The needed variable trajectory and variable pose kinematic characteristics of non-contact cutting tool are figured out and a metamorphic CNC flame cutting machine designed through metamorphic principle is presented. To analyze kinematic accuracy of the machine, models of joint clearances, manufacturing tolerances and errors in the input variables and error models considering the combined effects are derived based on screw theory after establishing ideal kinematic models. Numerical simulations, processing experiment and trajectory tracking experiment are conducted relative to an eccentric hole with bevels on cylindrical surface respectively. The results of cutting pass contour and kinematic error interval which the position error is from–0.975 mm to +0.628 mm and orientation error is from–0.01 rad to +0.01 rad indicate that the developed machine can complete cutting process continuously and effectively, and the established kinematic error models are effective although the interval is within a ‘large’ range. It also shows the matching property between metamorphic principle and variable working tasks, and the mapping correlation between original designing parameters and kinematic errors of machines. This research develops a metamorphic CNC flame cutting machine and establishes kinematic error models for accuracy analysis of machine tools.

Bending and vibrations analyses of laminated beams by using a zig-zag-layer-wise theory

Abstract This paper proposes one-dimensional layer-wise theories that make use of higher-order zig-zag functions defined over fictitious/mathematical layers of the cross-sectional area. These advanced kinematics enable the computational costs to be reduced while the accuracy of the classical layer-wise theories in which the number of physical and numerical layers coincide, is maintained. Variable kinematics theories have been obtained using piecewise continuous power series expansions of an arbitrary order defined over the whole cross-section of the structure. As in the classical layer-wise approach, the cross-section can be divided into a variable number of mathematical subdomains. The expansion order of each subdomain is therefore an input parameter of the analysis. This feature enables the solution to be refined locally as the kinematics expansion can be enriched over generic regions of the cross-section. The governing equations have been obtained by applying the Principle of Virtual Displacements, along with the Carrera Unified Formulation, and have been solved using the Finite Element method. Numerical simulations have been performed considering laminated and sandwich beams with very low length-to-depth ratio values. Comparisons between the present results and solutions available in the literature have pointed out the advantages of this approach, in terms of accuracy of the displacements, of the stress distributions over the beam cross-section and of the natural frequencies with respect to the classical layer-wise theories.

Assessment of recent zig-zag theories for laminated and sandwich structures

Abstract The flexural response of laminated composite and sandwich beams/plates under static distributed loading, classical and non-classical boundary conditions (simply-supported, cantilever and propped-cantilever) and geometric/constitutive heterogeneity of layers is analysed. As structural models, a recently developed equivalent single-layer zig-zag model by the authors, a discrete-layer model and a sublaminate model developed from it in this paper are used. Their contribution is to consider the continuity of the transverse normal stress and its through-thickness gradient at layer interfaces, as prescribed by the elasticity theory in addition to kinematic and transverse shear stress interlayer continuity customary considered in the literature. To this purpose, a piecewise variation of the three displacement components is adopted. The zig-zag amplitude expressions are obtained in closed-form from the enforcement of stress continuity conditions. To be refined without affecting costs, the equivalent single-layer model has variable kinematics and just five unknowns. Instead, sublaminate and discrete-layer models are refined by increasing the number of computational layers and variables. The aim of this paper is to assess whether the equivalent single-layer model having a computational cost comparable to that of classical models can be as accurate ad discrete-layer and sublaminate models. Benchmarks are presented, for which exact elasticity and approximate solutions are available for comparisons in literature. It is illustrated the utility of considering a variable kinematics for obtaining accurate stress predictions from constitutive equations and the transverse normal deformability for keeping equilibrium. The equivalent single-layer model is shown as accurate as discrete-layer and sublaminate models in all cases examined.

INDENTATION OF SANDWICHES USING A LAYERWISE MODEL WITH FIXED DEGREES OF FREEDOM

A zig-zag plate model with variable kinematics and fixed degrees of freedom recently developed by the authors is applied to study indentation of sandwiches with honeycomb/foam core, with the aim of reducing the computational burden. This model, which a priori fulfils the stresses and displacements continuity requirements, has a hierarchic representation of displacements, but just five degrees of freedom. The core crushing behaviour is determined apart once at a time in order to account for the variable elastic properties of the core during the structural analysis. Specifically, shell elements with elastic-plastic isotropic material are adopted to discretize the cellular structure and the faces, instead solid elements with nonlinear material behaviour are chosen to model foams. Once the properties of core are determined, the analysis is quickly carried out in closed form by the zig-zag model. As customarily, the onset of damage is determined using stress based criteria, while its growth is accurately and efficiently described using a mesoscale model. In this way, the degraded properties of the failed regions are evaluated with a physically based model, instead of guessing suited multiplication factors. The numerical results of simulations are shown to be in good agreement with experiments taken from the literature

Shell Finite Elements for the Analysis of Multifield Problems in Multilayered Composite Structures

This paper deals with the analysis of layered structures under thermal and electro-mechanical loads. Constitutive equations for multifield are considered and the Principle of Virtual Displacements (PVD) is employed to derive the governing equations. The MITC9 shell finite element based on the Carrera's Unified Formulation (CUF) has been applied for the analysis. The models grouped in the CUF have variable through-the-thickness kinematic and they provide an accurate distribution of displacements and stresses along the thickness of the laminate. The shell element has nine nodes and the Mixed Interpolation of Tensorial Components (MITC) method is used to contrast the membrane and shear locking phenomenon. The finite element analysis of multilayered plates and shells has been addressed. Variable kinematics, as well as layer-wise and equivalent single layer descriptions, have been considered for the presented FEs, according to CUF. A few problems are analyzed to show the effectiveness of the proposed approach. Various laminations, thickness ratios and curvature ratios are considered. The results, obtained with different theories contained in the CUF, are compared with both the elasticity solutions given in literature and the analytical solutions obtained using the CUF and the Navier's method.

Thermal stress analysis of laminated structures by a variable kinematic MITC9 shell element

ABSTRACTA linear static thermal stress analysis of composite shell structures is carried out by means of a shell finite element with variable through-the-thickness kinematic. The refined models used are both Equivalent Single Layer (ESL) and Layer Wise (LW) and they are grouped in the Unified Formulation by Carrera (CUF). These models permit the distribution of displacements, stresses and temperature along the thickness of the multilayered shell to be accurately described. The Principle of Virtual Displacement (PVD) is employed to derive the governing equations. The Mixed Interpolation of Tensorial Components (MITC) method is used to contrast the membrane and shear locking phenomenon for a nine-node shell element. Cross-ply plate, cylindrical and spherical shells with simply supported edges and subjected to bi-sinusoidal thermal load are analyzed and various thickness ratios are considered. The results, obtained with different theories contained in the CUF, are compared with both the elasticity solutions given in the literature and the analytical solutions obtained using higher-order models and Navier's method. From the analysis, one can conclude that the shell element based on the CUF is very efficient, and its use leads to reach higher accuracy than classical models in the study of layered structures.

A Sublaminate Generalized Unified Formulation for the analysis of composite structures

This paper presents a very flexible variable kinematics modeling technique for composite structures. The key point is the subdivision of the multilayered cross-section into numerical sublaminates, each consisting of one or more physical plies, and the formulation of model assumptions independently in each sublaminate according to the compact index notation known as Generalized Unified Formulation. Reference is made to the classical displacement-based and the mixed RMVT variational frameworks. For each sublaminate, the user may thus freely choose the order of the through-thickness polynomial distribution of each unknown as well as whether the unknown is described in an Equivalent Single Layer (ESL) or Layer-Wise (LW) approach. Sublaminates are always assembled in a Layer-Wise manner. This Sublaminate-GUF (S-GUF) is generally applicable to multilayered structures, but is particularly useful for sandwich panels, in which different models may be used for the thick, soft core and the thin, stiff skins. A Navier-type solution is used for demonstrating the validity of this approach by means of benchmark problems concerning the static response of sandwich plates.

Benchmark solutions and assessment of variable kinematics models for global and local buckling of sandwich struts

This paper investigates the periodic global buckling and wrinkling of sandwich struts subjected to uniaxial loads. An exact elasticity solution is obtained in the plane strain setting for selected benchmark problems and serves as reference for a systematic assessment of several displacement-based sandwich plate models. The Sublaminate Generalized Unified Formulation is used to generate the algebraic two-dimensional governing equations of the variable kinematics models, for which a Navier-type solution is adopted.

Reissner's Mixed Variational Theorem and variable kinematics in the modelling of laminated composite and FGM doubly-curved shells

The present article proposes a mixed displacements/transverse stresses approach for the free vibration analysis of laminated composite and FGM doubly-curved shells. The theoretical formulation is derived by combining Reissner's Mixed Variational Theorem (RMVT), Carrera's Unified Formulation (CUF) and the Ritz method. With the application of the RMVT the interlaminar equilibrium of the transverse normal and shear stresses is fulfilled a priori by exploiting the use of Lagrange multipliers. The transverse normal and shear stresses become primary variables within the formulation and are modelled with a Layer-Wise (LW) kinematics description. However, on the other hand, displacement variables, which describe the kinematics of the shell structures, are defined using Equivalent Single Layer (ESL), Zig-Zag (ZZ) and LW shell theories. The Mixed Hierarchical Trigonometric Ritz Formulation (MHTRF) is then used as solution technique to compute the natural frequencies of laminated composite and FGM doubly-curved shells. Several study-cases are addressed and the proposed RMVT-based shell models are assessed by comparison with both 3D elasticity and 3D Ritz solutions. The effect of significant parameters such as orthotropic ratio, stacking sequence, aspect ratio, lamination angle, length-to-thickness and radius-to-length ratios as well as volume fraction index on the dimensionless frequency parameters is discussed.

Accurate Response of Wing Structures to Free-Vibration, Load Factors, and Nonstructural Masses

Based on the Carrera unified formulation, this work extends variable kinematic finite beam elements to include load factors and nonstructural masses for the static and vibration analyses of complex, metallic wing structures. According to the Carrera unified formulation, variable kinematic beam theories are formulated in an automatic and hierarchical manner by expressing the displacement field as an arbitrary expansion through generic cross-sectional functions. Both Taylor-like and Lagrange polynomials are used in this paper to develop refined beam kinematics, and the related theories are referred to as Taylor expansion and Lagrange expansion, respectively. The generalized unknowns of Taylor expansion models are the beam axis displacements and the -order displacement derivatives, with being a free parameter of the analysis. Classical beam theories are clearly particular cases of the linear () Taylor expansion model. On the other hand, Lagrange expansion models have only pure translational displacements as unknowns. By exploiting this characteristic of Lagrange expansion, a componentwise approach is implemented and used for the analysis of multicomponent reinforced-shell structures. Numerical applications are developed by classical finite element procedures, and both static response and free vibration analyses are addressed. Various configurations of a benchmark wing are considered, and the capabilities of the present methodologies when dealing with higher-order effects due to deformable cross sections and geometrical discontinuities (for example, underside windows) are evaluated. The attention is focused on the applicability of the present refined beam models to problems involving complex, external inertial loadings. The results are compared to finite element solutions from commercial tools, including full three-dimensional models and models obtained by assembling two-dimensional shell and one-dimensional finite elements.

Accurate Free Vibration Analysis of Launcher Structures Using Refined 1D Models

This work uses different finite element approaches to the free vibration analysis of reinforced shell structures, and a simplified model of a typical launcher with two boosters is used as an example. The results obtained using a refined one-dimensional (1D) beam model are compared to those obtained with commercial finite element software. The 1D models that are used in the present work are based on the Carrera Unified Formulation (CUF), which assumes a variable kinematic displacement field over the cross-sections of the beam. Two different sets of polynomials that correspond to Taylor (TE) or Lagrange (LE) expansions were used. The analyses focused on three reinforced structures: a stiffened panel, a reinforced cylinder and the complete structure of the launcher. The frequencies and natural modes obtained using one-dimensional models are compared to those obtained from classical finite element analysis. The classical FE models were built using a beam-shell or solid elements, and the results indicate that the refined beam models can in fact be used to investigate the behavior of very complex reinforced structures. These models can predict the shell-like modes that are typical of thin-walled structures that cannot be detected using classical beam models. The refined 1D models used in the present work provide results that are as accurate as those from solid FE models, but the 1D models have a much lower computational cost.

프로골퍼의 드라이버 비거리증가를 위한 목적스윙 시 X-Factor, X-Factor Stretch, 클럽변인과 전체비거리(total length)와의 상관관계

프로골퍼의 드라이버 비거리증가를 위한 목적스윙 시 X-Factor, X-Factor Stretch, 클럽변인과 전체비거리(total length)와의 상관관계

Evaluation of mixed theories for laminated plates through the axiomatic/asymptotic method

This paper proposes variable kinematic, mixed theories for laminated plates built via the asymptotic/axiomatic method (AAM). This method has been recently developed and successfully applied to develop refined theories for multilayered plates and shells. The AAM evaluates the accuracy of each unknown variables of a structural model. The present paper extends the AAM to mixed theories based on the Reissner Mixed Variational Theorem (RMVT). The displacement and transverse stress fields are modeled by means of the Carrera Unified Formulation (CUF), and expansions up to the fourth-order are employed. Equivalent Single Layer (ESL) and Layer Wise (LW) schemes are adopted, and closed-form Navier-type solutions are considered.The AAM is exploited to determine the set of active terms of a refined plate model. The inactive terms are then discarded. The effectiveness of each variable is evaluated with respect to an LW, fourth-order mixed model. Reduced models are built for different thickness ratios, stacking sequences and displacement/stress variables.The results suggest that reduced models with significantly less unknown variables than full models can be built with no accuracies penalties. Such models are problem dependent, and full models should be preferred in the case of thick, asymmetric plates.

Heat conduction and Thermal Stress Analysis of laminated composites by a variable kinematic MITC9 shell element

Abstract The present paper considers the linear static thermal stress analysis of composite structures by means of a shell finite element with variable through-thethickness kinematic. The temperature profile along the thickness direction is calculated by solving the Fourier heat conduction equation. The refined models considered are both Equivalent Single Layer (ESL) and Layer Wise (LW) and are grouped in the Unified Formulation by Carrera (CUF). These permit the distribution of displacements, stresses along the thickness of the multilayered shell to be accurately described. The shell element has nine nodes, and the Mixed Interpolation of Tensorial Components (MITC) method is used to contrast the membrane and shear locking phenomenon. The governing equations are derived from the Principle of Virtual Displacement (PVD). Cross-ply plate, cylindrical and spherical shells with simply-supported edges and subjected to bi-sinusoidal thermal load are analyzed.Various thickness ratios and curvature ratios are considered. The results, obtained with different theories contained in the CUF, are compared with both the elasticity solutions given in the literature and the analytical solutions obtained using the CUF and the Navier’s method. Finally, plates and shells with different lamination and boundary conditions are analyzed using high-order theories in order to provide FEM benchmark solutions.

Variable kinematics models and finite elements for nonlinear analysis of multilayered smart plates

A variable kinematics approach for moderately large deflection analysis of smart magneto-electro-elastic multilayered plates is presented. The approach is based on the condensation of the electro-magnetic state into the plate kinematics, whose nonlinear strain–displacement relationships are expressed in the von Karman sense. This leads to models resulting in an effective mechanical plate, which takes the multifield coupling effects into account by the plate stiffness, inertia and loading characteristics, consistently defined as combinations of the layers material properties. By a unified approach, both equivalent single layer and layerwise models are developed formulating the associated isoparametric finite elements. Results are presented for different kinematics with through-the-thickness expansion up to the fourth order with the aim of validating the modeling strategy and showing its features for the investigation of smart plates geometrical nonlinear behaviour.

Comparison of various 1D, 2D and 3D FE models for the analysis of thin-walled box with transverse ribs subjected to load factors

This paper evaluates various Finite Element (FE) models for the static and dynamic analyses of single- and multi-bay metallic box structures. Load factors are considered in the static response, whereas free vibration analyses are addressed to compare the dynamic performances. Different FE models able to include cross-sectional deformations are compared. These comprise solid (3D) and plate/shell (2D) models, which are obtained by using a general-purpose commercial software. Results related to a hierarchical variable kinematic beam (1D) formulation, which opportunely degenerates into classical beam theories with rigid-cross section assumptions (e.g. Euler–Bernoulli and Timoshenko), are also addressed. Displacements as well as axial and shear stress fields are compared for various loading cases. Regarding dynamic analyses, fundamental and higher-order natural frequencies by various models are computed and the Modal Assurance Criterion is used to compare the eigenmodes. It is concluded that refined models are mandatory to accurately describe the static and dynamic characteristics of thin-walled box structures. Classical and lower-order refined beam models show severe limitations in capturing localized stress/strain fields as well as local mode shapes. Nevertheless, the accuracy of lower-order models is improved as a consequence of the adoption for transverse ribs, which lead to more rigid cross-sections.