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Copyright of Ordnance Industry Automation is the property of Editorial Board for Ordnance Industry Automation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Background: The estimation of the natural frequency is a key step in the design of an offshore wind turbine (OWT). In the literature, the natural frequency can be evaluated through two steps, the first one is the estimation of the fixed base natural frequency using formulas from the literature, and the second one is the estimation of the foundation system effect. These formulas are efficient for simple shapes of OWTs but become hard to use for recent OWTs. Method: The finite element method and the Fourier series have been used to develop a computer code under the name of "TurbiSoft" to compute the natural frequency of the OWT with the foundation system. Each of the linear, the quadratic volume elements, and the joint element was employed in TurbiSoft to model the whole system to examine the influence of the element type, the soil profile, and the interaction soil–pile state on the natural frequency. Results: The outcomes of TurbiSoft for several OWTs were compared with the measured frequencies to define the range error of each parameter. The results show that the quadratic elements are more suitable for the natural frequency calculation. Also, this study demonstrates that the natural frequency value is strongly related to the interaction soil–pile state, and is slightly affected by the soil profile variation. [ABSTRACT FROM AUTHOR]

We consider an additive Vanka‐type smoother for the Poisson equation discretized by the standard finite difference centered scheme. Using local Fourier analysis, we derive analytical formulas for the optimal smoothing factors for vertex‐wise and element‐wise Vanka smoothers. In one dimension the element‐wise Vanka smoother is equivalent to the scaled mass operator obtained from the linear finite element method and in two dimensions the element‐wise Vanka smoother is equivalent to the scaled mass operator discretized by bilinear finite element method plus a scaled identity operator. Based on these findings, the mass matrix obtained from finite element method can be used as a smoother for the Poisson equation, and the resulting mass‐based relaxation scheme yields small smoothing factors in one, two, and three dimensions, while avoiding the need to compute an inverse of a matrix. Our analysis may help better understand the smoothing properties of additive Vanka approaches and develop fast solvers for numerical solutions of other partial differential equations. [ABSTRACT FROM AUTHOR]

The use of splines to model the geometry and perform iso-geometric analysis is fast replacing conventional finite element analysis. In this paper, the main objective is to analyze the equation of motion using basis splines. The equation of motion is a function of both space and time. The finite difference method is used to evaluate the newer values of the function at each time step. The splines are used to determine the mass matrix and the stiffness matrix. The sample code is also presented. The displacement vector at each time step is shown using graphs. The graphs clearly show how the member vibrates at each time step and the state of the beam. The analysis clearly shows how the Galerkin method can be applied with a weak formulation to solve for the displacement vector in a step-by-step method. Two problems are discussed, the first is a classroom example, and the second is a home work example, which may require a bit of effort to determine the value of the nodal displacement vector at several nodes for each time step. [ABSTRACT FROM AUTHOR]

The design of multi-storey buildings is a natural trend in the development of a modern metropolis. Obtaining exact solutions when studying their own and forced oscillations within the framework of a continuous homogeneous medium model (continuum mechanics) with an infinite number of degrees of freedom is often difficult to implement. Therefore, in the article (as part of the modernization of the finite element method), the model of a multi-storey building is discretized and endowed with a finite number of degrees of freedom placed in the middle of the finite elements at the nodes (the mass of finite elements is also placed there), which elastically interact with the finite elements of the model that do not have mass. It is believed that the elements of a multistorey building work only for bending, which is fully justified by comparing the frequencies of its bending and longitudinal oscillations. The resolving system of differential equations of oscillations of a multi-storey building, in which expressions for energies (potential, kinetic and Rayleigh) are written in quadratures, is obtained using Lagrange equations of the second kind. In the article, the problems of free oscillations of 3- and 100-storey buildings are solved using Green’s functions, stiffness, mass, compliance matrices, etc. The approximate results obtained in the article, when compared with the little-known approximate results obtained by other methods, as well as exact results (direct and indirect methods of boundary elements), showed a good correspondence.

The design of multi-storey buildings is a natural trend in the development of a modern metropolis. Obtaining exact solutions when studying their own and forced oscillations within the framework of a continuous homogeneous medium model (continuum mechanics) with an infinite number of degrees of freedom is often difficult to implement. Therefore, in the article (as part of the modernization of the finite element method), the model of a multi-storey building is discretized and endowed with a finite number of degrees of freedom placed in the middle of the finite elements at the nodes (the mass of finite elements is also placed there), which elastically interact with the finite elements of the model that do not have mass. It is believed that the elements of a multistorey building work only for bending, which is fully justified by comparing the frequencies of its bending and longitudinal oscillations. The resolving system of differential equations of oscillations of a multi-storey building, in which expressions for energies (potential, kinetic and Rayleigh) are written in quadratures, is obtained using Lagrange equations of the second kind. In the article, the problems of free oscillations of 3- and 100-storey buildings are solved using Green’s functions, stiffness, mass, compliance matrices, etc. The approximate results obtained in the article, when compared with the little-known approximate results obtained by other methods, as well as exact results (direct and indirect methods of boundary elements), showed a good correspondence.

Copyright of Engineering Mechanics / Gongcheng Lixue is the property of Engineering Mechanics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Proper use of materials is one of the most important criteria of a rational design and shaping of engineering constructions. It requires such dimensioning of each element of the construction which will ensure that the element is matched to its load - and this condition is fulfilled only for beams with variable cross section. Hence, it is essential to develop possibilities of calculations of beams with cross section varying along the beam longitudinal axis. This study provides relevant matrices (i.e. stiffness, mass and initial stress matrix) applied in the Finite Element Method for calculations of natural frequencies and buckling critical forces. The matrices have been derived for beams shaped as a truncated cone with a linear generatrix, supported in various ways. The results have been compared to those obtained for the stair-shaped beams approximating the conical ones; a good concordance of results has been stated. [ABSTRACT FROM AUTHOR]

Assessment of the natural frequency of the offshore wind turbine (OWT) system is a critical task in design to avoid resonance. The natural frequency of the monopile supported OWT may change during operation and needs to be calculated regularly. Conventional numerical methods separately model different components of the OWT system using various element types, which require tedious data manipulations and are inefficient to assess the changing natural frequency of the monopile supported OWTs. This paper develops a unified beam-column element by directly integrating the soil-pile interactions in element formulation. The proposed method is applicable for different components of the OWT system, including the rotor, the tower, the transition piece, and the monopile foundation, which enables an integrated and efficient OWT analysis by using only one type of element. The pile-soil interactions are directly considered in the element formulation. The tedious soil spring elements are avoided. Definitions and formulations of the proposed element are provided and the numerical solution procedure for the natural frequency analysis of OWTs is illustrated. The accuracy of the proposed element is validated against several closed-form solutions and the distributed spring model. The proposed method is also applied to analyze the natural frequencies of six OWTs in practice. The calculated results are consistent with the field measured data. [ABSTRACT FROM AUTHOR]

One important aspect of the dynamic performance evaluation of a parallel manipulator is the elastodynamic analysis which calls for the calculation of the stiffness and mass matrices. In the elastodynamic analysis, the motors are assumed to be locked, and thus, the system's mass matrix should be obtained based on the small-amplitude elastic motion caused by the flexible components. As a sequel to the efficient methodology for the calculation of the stiffness matrix of a multibody system which was introduced in our previous work, the focus of this article is to develop a similar mathematical framework to compute the mass matrix of a rigid-flexible multibody system parametrically. This similarity helps an analyst to readily obtain the kinetic and strain energies with the same set of generalized coordinates, and consequently, the mass and stiffness matrices with the same dimensions. Therefore, the elastodynamic analysis of complicated multibody systems can be conducted with no burden. Moreover, the resultant mass matrix can be obtained parametrically which can be used for optimization and sensitivity analysis purposes. The proposed framework is applied for the elastodynamic analysis of a Delta parallel robot, and the results will be verified by means of a finite element analysis software package. [ABSTRACT FROM AUTHOR]

Free vibration analyses of plate on elastic foundations are very common in mechanical and structural engineering and these types of structures are analysed using the concept of "plates on elastic foundation". It is very important to build more practical and simple foundation models for safe and economical design of transversely loaded plate. In the present study, a workable approach for vibration analysis of laterally loaded plate on Pasternak foundation and second-order (C²) compatible plate bending elements based on the Kirchhoff theory is attempted. A Matlab code is developed for the present formulation. The results, thus obtained, are compared with similar studies done by other researchers, which show very good conformity. Parametric studies are carried out to obtain response for different soil parameters and boundary conditions. It is concluded that the influence of the Winkler coefficient on the frequency parameters are larger than that of the aspect ratio and Pasternak shear coefficient regardless of boundary conditions. The present approach is simple, effective and more reliable for vibration analysis of plates on Pasternak foundation and the convergence rate is very high and takes less computational efforts, resources and times. [ABSTRACT FROM AUTHOR]

• Unifying six beam theories into a single equation. • Unified dynamic stiffness matrix and transfer matrix. • Frequency-dependent mass and stiffness matrices. • Thorough comparison and scope of different beam theories for curved beams. High-precision, efficient and streamlined methods for the dynamic modeling and analyzing of engineering structures have consistently captured attention of designers and researchers. This paper addressed this increasing demand by presenting a generalized modeling and analyzing method. Specially, a unified model was proposed to integrate six beam theories into a single formula. Upon this model, a unified dynamic stiffness matrix was derived, serving as an elemental building block for formulating the transfer matrix, frequency-dependent mass and stiffness matrices as well as further integrating finite element method, dynamic stiffness matrix method and transfer matrix method. Through this transition, challenges associated with transcendental eigenvalue in dynamic stiffness matrix method and transfer matrix method was effectively resolved and the advantages of transfer matrix method in handling complex structure were maximized. Given the widespread utilization of curved beams in buildings, bridges, robotics and sensors, the proposed method was validated through comparative analysis with other methods, focusing on both curved and straight beams. Furthermore, this study delineated applicability scopes of different beam theories and provided the strategic approach for curved and straight beams. The feasibility of this strategy was demonstrated through an investigation involving a corrugated structure comprising both straight and curved beams. [ABSTRACT FROM AUTHOR]

This paper deals with the fast solution of linear systems associated with the mass matrix, in the context of isogeometric analysis. We propose a preconditioner that is both efficient and easy to implement, based on a diagonal-scaled Kronecker product of univariate parametric mass matrices. Its application is faster than a matrix–vector product involving the mass matrix itself. We prove that the condition number of the preconditioned matrix converges to 1 as the mesh size is reduced, that is, the preconditioner is asymptotically equivalent to the exact inverse. Moreover, we give numerical evidence of its good behaviour with respect to the spline degree and the (possibly singular) geometry parametrization. We also extend the preconditioner to the multipatch case through an Additive Schwarz method. [ABSTRACT FROM AUTHOR]

This paper presents a new co-rotational beam element based on quaternion algebra as parametrization of large rotations for dynamic applications. The co-rotational framework is based on a decomposition of the kinematics into a rigid element frame (that follows the element) and its pure deformation. Based on a previous work related to the static equilibrium, the proposed co-rotational element lies in the framework of the incremental formulations. The kinematic field description using quaternion algebra allows to calculate quaternion's velocities and accelerations. Then, the dynamic equilibrium and the formulation of the internal mass matrix and gyroscopic terms are detailed. Seven numerical applications in dynamic conditions are presented and compared with literature results showing the response of the proposed element in terms of conservation of the energy and convergence. • Quaternion parametrization of large rotations in a dynamic co-rotational framework. • Dynamic operators are obtained using quaternion's velocities and accelerations. • Linearization of the response to obtain a dynamic algorithmic tangent operator. • Seven distinct dynamic study cases are presented showing numerical capabilities. [ABSTRACT FROM AUTHOR]

Assessment of the natural frequency of the offshore wind turbine (OWT) system is a critical task in design to avoid resonance. The natural frequency of the monopile supported OWT may change during operation and needs to be calculated regularly. Conventional numerical methods separately model different components of the OWT system using various element types, which require tedious data manipulations and are inefficient to assess the changing natural frequency of the monopile supported OWTs. This paper develops a unified beam-column element by directly integrating the soil-pile interactions in element formulation. The proposed method is applicable for different components of the OWT system, including the rotor, the tower, the transition piece, and the monopile foundation, which enables an integrated and efficient OWT analysis by using only one type of element. The pile-soil interactions are directly considered in the element formulation. The tedious soil spring elements are avoided. Definitions and formulations of the proposed element are provided and the numerical solution procedure for the natural frequency analysis of OWTs is illustrated. The accuracy of the proposed element is validated against several closed-form solutions and the distributed spring model. The proposed method is also applied to analyze the natural frequencies of six OWTs in practice. The calculated results are consistent with the field measured data.

The effect of the nonlocal parameter on the free vibration analysis of nano scaled trusses and frames is examined. Accordingly, firstly, the axial and bending vibrations of the nano scaled longitudinal element are formulated. Simple rod and Euler-Bernoulli assumptions are considered for axial and bending vibrations, respectively. Finite element matrices are obtained by applying the average weighted residue to the nonlocal equation of motion for free vibration. These matrices are combined according to the freedoms of longitudinal element and the matrix displacement method is explained for structures consisting of discrete longitudinal elements. It is discussed how the classical stiffness and mass matrices are modified by the atomic parameter. [ABSTRACT FROM AUTHOR]

A preconditioner for the mass matrix for high order finite element discretization on tetrahedra is presented and shown to give a condition number that is independent of both the mesh size and the polynomial order of the elements. The preconditioner is described in terms of a new, high order basis which has the usual property whereby individual functions are associated with distinct geometric entities of the tetrahedron. It is shown that the basis enjoys the novel property that the resulting mass matrix is spectrally equivalent to its own diagonal with constants independent of h and p. Although the exposition is based on an explicit basis, the preconditioner can be applied to any choice of basis. In particular, the basis can be used to specify a basis-independent additive Schwarz method, meaning that, in order to apply the preconditioner to an alternative basis, one only need to implement an appropriate change of basis. [ABSTRACT FROM AUTHOR]

Different inertia representations can lead to different formulations of the differential-algebraic equations for the quaternion-based rigid body dynamics. In this paper, the inertia representations are classified into α -type and γ -type, according to the additional parameters in the kinetic energy. These two types of representations and the corresponding parameters α and γ are theoretically equivalent if the constraint q T q = 1 is satisfied exactly. Nevertheless, the error estimation demonstrates that they can present entirely different numerical features in simulation and suggests that the parameter γ can be used to optimize the numerical performance of the integrations in simulation. To further verify the numerical difference between the inertia representations of α -type and γ -type, the corresponding modified Hamilton's equations are discretized by the IMS (implicit midpoint scheme), EMS (energy–momentum preserving scheme) and Gauss–Lobatto SPARK methods. Numerical performance for the examples of the spinning symmetrical top is shown to result from the comprehensive effect of the discretization schemes including the distribution of discretized points and the convergence order, the inertia representations and their combinations. Numerical results further suggest that the integrations of γ -type are superior to those of α -type and the optimized values of γ can be used to achieve better numerical accuracy, convergence speed and stability. [ABSTRACT FROM AUTHOR]

Functionally graded material (FGM) is a substitute for laminated composite materials. It avoids the disadvantages of laminated composites. Unlike laminated composite materials, in FGM, there is a continuous change in properties across the depth. It is widely used in the areas of aerospace, biomechanics, automobile, etc. Hence, static and dynamic analysis of FGM is of interest today. Researchers have developed analytical methods for both static and dynamic analyses of FGM. But, numerical analysis is essential to avoid assumptions in the analysis. Usually, the most popular numerical method, the finite element method (FEM), is used for the analysis of FGM structures. To implement the variation in properties in finite element software, FGM is thoroughly meshed along the direction of variation in property. The applied element method (AEM) developed in 1997 has some advantages when compared to FEM. Faster processing, lesser memory requirement and simplicity of analysis are some of them. Usually, most of the research on AEM is on concrete and steel structures. From the formulation, it can be seen that AEM can be used most effectively for FGM. So, in this paper, this aspect is highlighted by conducting static analysis and solving eigen value problem in dynamic analysis of beams made of FGM. The approach in conventional AEM to determine stiffness matrix is simplified so that it is suitable for the analysis of functionally graded (FG) beams. Two-dimensional elements are developed for FG beams with various material property distribution laws. The stiffness matrix and mass matrix are derived, and the method to find strain and stress at various points in FG beams is discussed. The deflection and stress distribution obtained by AEM is compared with those determined by analytical means. Finite element analysis is also carried out and compared. The natural frequency of beams with different length-to-depth ratio and support conditions are found and compared with the results in the literature. From the study, it is seen that AEM can be effectively used for the analysis of structures made of FGM. [ABSTRACT FROM AUTHOR]

For molecules with constraints such as fixed lengths and angles, it is more efficient to consider the molecular movement in the space of generalised internal coordinate than in Cartesian coordinate. This paper presents a new framework in the simulation of molecular movement, especially for macro-molecules with massive length and angle constraints. The generalised forces are calculated to invert the dense mass matrix for integrating the constraints equation. The inverting of the mass matrix was made based on distance descending ordering method. The method does an reordering of the internal variables to make the Cholesky decomposition need no fill-in, which promises the O (n) time complexity in doing mass matrix inverting. The method was extended for application to loop structures. It is found that the mass matrix would become singular when the bond angles approach 0 or π; thus a rotation convention switch method was proposed to resolve the singularity. The O (n) time complexity has been demonstrated and the length and angle constraints can be arbitrarily applied. The long-time energy conservation in NVE ensemble was compared for results from symplectic and non-symplectic time integrators. The non-symplectic fourth-order Runge–Kutta method still has satisfactory long-time energy conservation if using small time step. [ABSTRACT FROM AUTHOR]

Abstract Neutrino mass sum rules have recently gained again more attention as a powerful tool to discriminate and test various flavour models in the near future. A related question which has not yet been discussed fully satisfactorily was the origin of these sum rules and if they are related to any residual or accidental symmetry. We will address this open issue here systematically and find previous statements confirmed. Namely, the sum rules are not related to any enhanced symmetry of the Lagrangian after family symmetry breaking but they are simply the result of a reduction of free parameters due to skillful model building.

The floating frame of reference formulation is an established method for the description of linear elastic bodies within multibody dynamics. An exact derivation leads to rather complex equations of motion. In order to reduce the computational burden, it is common to neglect certain terms. In the literature this is done by strict application of the small deformation assumption to the kinetic energy. This leads to a remarkably simplified set of equations. In this work, the significance of all terms is investigated at the level of the equations of motion. It is shown that for a large number of applications the previously mentioned set of simple equations is sufficient. Furthermore, scenarios are described in which this simple set is no longer accurate enough. Finally, guidelines are provided, so that engineers can decide which terms should be considered or not. The theoretical conclusions drawn in this work are underlined by qualitative numerical investigations. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Vibration Engineering is the property of Nanjing University of Aeronautics & Astronautics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)