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In this paper, the numerical properties of a recently developed high‐order Spectral Euler‐Bernoulli Beam Element (SBE) featuring a C1‐continuous approximation of the displacement field are assessed. The C1‐continuous shape functions are based on two main ingredients, which are an Hermitian interpolation scheme and the use of Gauß‐Lobatto‐Legendre (GLL) points. Employing GLL‐points does not only avoid Runge oscillations, but also yields a diagonal mass matrix when exploiting the nodal quadrature technique as a mass lumping scheme. Especially in high‐frequency transient analyses, where often explicit time integration schemes are utilized, having a diagonal mass matrix is an attractive property of the proposed element formulation. This is, however, achieved at the cost of an under‐integration of the mass matrix. Therefore, a special focus of this paper is placed on the evaluation of the numerical properties, such as the conditioning of the element matrices and the attainable rates of convergence (ROCs). To this end, the numerical behavior of the SBEs is comprehensively analyzed by means of selected benchmark examples. In a nutshell, the obtained results demonstrate that the element yields good accuracy in combination with an increased efficiency for structural dynamics exploiting the diagonal structure of the mass matrix. [ABSTRACT FROM AUTHOR]

Modal analysis of the Reynolds‐Orr energy equation for a miscible viscosity stratified slippery channel flow is deliberated. The main aim is to extend the earlier work of Ghosh etal. (Phys. Fluid, Vol. 26, 014107 (2014)) to discusses the instability mechanism, which has not been investigated so far. The generalized equation governing the average rate of change of disturbance kinetic energy is evaluated after solving the Orr‐Sommerfeld boundary value problem. The analysis includes viscosity perturbation and slip boundary condition. Maximum growth rate curves for two‐dimensional disturbances reconfirm the existence of new unstable modes at low Reynolds numbers. Stabilizing and destabilizing effects of wall velocity slip are found depending on parameter regime. The physical mechanism responsible for the dual role of the wall slip is brought to light by estimating the alteration of kinetic energy due to the production and dissipation of energy originating from various factors inside the perturbed flow. Slip boundary condition at the wall affects the base flow as well as perturbations. Our numerical calculations here show how the wall velocity slip gives impact on each energy terms. The results ensure that, the new unstable mode is occurred due to the overlap of velocity and viscosity disturbance vortices. Stability characteristics of this overlap instability are ruled by the energy sourcing from the Reynolds stress and the viscosity perturbation gradient effect. Modal analysis of the Reynolds‐Orr energy equation for a miscible viscosity stratified slippery channel flow is deliberated. The main aim is to extend the earlier work of Ghosh etal. (Phys. Fluid, Vol. 26, 014107 (2014)) to discusses the instability mechanism, which has not been investigated so far. The generalized equation governing the average rate of change of disturbance kinetic energy is evaluated after solving the Orr‐Sommerfeld boundary value problem. The analysis includes viscosity perturbation and slip boundary condition. Maximum growth rate curves for two‐dimensional disturbances reconfirm the existence of new unstable modes at low Reynolds numbers. Stabilizing and destabilizing effects of wall velocity slip are found depending on parameter regime... [ABSTRACT FROM AUTHOR]

Gustav Kirchhoff has been credited, among many other renowned achievements, as the first scientist who tackled and solved the problem of studying the transversal vibrations of beams with variable cross-section. His contribution, which was presented in 1879 and published in the following year, is nowadays almost forgotten in the international scientific community, with the only exception of the German-speaking countries. For this reason it is rediscovered and thoroughly discussed here, with an exegetical approach. For completeness' sake an unabridged translation into English (the first one, to the best of the authors' knowledge) is provided in the appendix for the interested readers. [ABSTRACT FROM AUTHOR]

The theoretical results relevant to the vibration modes of Timoshenko beams are here used as benchmarks for assessing the correctness of the numerical values provided by several finite element models, based on either the traditional Lagrangian interpolation or on the recently developed isogeometric approach. Comparison of results is performed on both spectrum error (in terms of the detected natural frequencies) and on the l2 relative error (in terms of the computed eigenmodes): this double check allows detecting for each finite element model, and for a discretization based on the same number of degrees-of-freedom, N, the frequency threshold above which some prescribed accuracy level is lost, and results become more and more unreliable. Hence a quantitative way of measuring the finite element performance in modeling a Timoshenko beam is proposed. The use of Fast Fourier Transform is finally employed, for a selected set of vibration modes, to explain the reasons of the accuracy decay, mostly linked to a poor separation of the natural frequencies in the spectrum, which is responsible of some aliasing of modes. [ABSTRACT FROM AUTHOR]

The damage acting on a structure can lead to disproportionate consequences, i.e., the global collapse. This extreme situation has to be avoided and, thus, structural monitoring is requested in those structures where human losses are possible and large economic consequences are expected. Static measurement devices are the most economic instrumental set-ups able to highlight the presence of progressive damages. However, this monitoring system suffers from the structural behaviour under the external loads. In many situations, alternate load paths shown the non-effectiveness of the measurement system since the instrumentation are installed on elements not relevant for the response under the given loads. The same problems occur when the structure is compartmentalized, i.e. the structural responses of the single parts dependent on the loads acting almost only on the same single component. In order to measure the degree of compartmentalization, different novel metrics based on stiffness matrix properties are proposed and their effectiveness discussed. The new idea of this paper is to connect compartmentalization of structures with a sort of distance of the stiffness matrix from the set of diagonal matrices. Few examples are illustrated. [ABSTRACT FROM AUTHOR]

A general method for the modal decomposition of the equations of motion of damped multi-degree-of-freedom-systems is presented. The first variant of the presented method in earlier publications of the author, including the complex right eigenvectors, is briefly reviewed first. The second presented variant is also based on the corresponding eigenvalue problem of the damped structure including the complex left and right eigenvectors. After initial partitioning of the equations of motion a real modal transformation matrix is built by a combination of two complex transformations. For the general case of damped structures with non-modal symmetric damping matrix a modal analysis can be performed in real arithmetic. Two numerical examples with 3 and 10 DOF's demonstrate the accuracy and the advantages of the presented modal solution method. [ABSTRACT FROM AUTHOR]

A direct time integration method is presented for the solution of the equations of motion describing the dynamic response of structural linear and nonlinear multi-degree-of-freedom systems. It applies also to large systems of second order differential equations with fully populated, non symmetric coefficient matrices as well as to equations with variable coefficients. The proposed method is based on the concept of the analog equation, which converts the coupled N equations into a set of single term uncoupled second order ordinary quasi-static differential equations under appropriate fictitious loads, unknown in the first instance. The fictitious loads are established from the integral representation of the solution of the substitute single term equations. The method is simple to implement. It is self starting, unconditionally stable and accurate and conserves energy. It performs well when large deformations and long time durations are considered and it can be used as a practical method for integration of the equations of motion in cases where widely used time integration procedures, e.g. Newmark's, become unstable. Several examples are presented, which demonstrate the efficiency of the method. The method can be straightforward extended to evolution equations of order higher than two. [ABSTRACT FROM AUTHOR]

The classical continuum model does not describe the mechanical behaviour of cellular materials. To model the size-dependent mechanical properties of these materials, the extended continuum models such as the micropolar approach can be used. The reference data is extracted from the virtual experiments performed on an artificial open-cell foam specimens. With the proper set of the Cosserat parameters the detailed microstructural model can be described on the macroscopic level using the extended continuum approach. The parameter identification task is defined as a minimisation task, which is solved using the evolutionary computational framework. [ABSTRACT FROM AUTHOR]

It is interesting to ask if there are any fundamental differences between Eulerian and Lagrangian formulations in structural dynamics problems. Steady rotating annuli formulated in both Eulerian and Lagrangian coordinates are exemplified for the present study. Nonlinear strain-displacement relationships for radial vibration only without circumferential motion in polar coordinates are derived. Linear and nonlinear strain-displacement relationships are implemented to form the governing equations and are compared to check for the validity of small strain analysis. By including the Poisson's effect, the resulting governing equations are nonlinear equations with material density redistribution and plate thickness variation. Analytical solutions of the linearized versions of nonlinear Eulerian and Lagrangian governing equations by keeping material density and plate thickness constant are offered. Fundamental issues in existence and uniqueness of analytical solutions are discussed between linear Eulerian and linear Lagrangian formulations. Furthermore, the fundamental difference of imposing free edge boundary conditions in Eulerian and Lagrangian coordinates is emphasized. Results show that there are in general insignificant differences between Eulerian and Lagrangian descriptions for circular plate problems, although the approaches to solve the problems in the two formulations are quite different. [ABSTRACT FROM AUTHOR]

The family of Newmark and generalized α-methods is extended to constrained mechanical systems by using simultaneous position and velocity stabilization as key ideas. In this way, the acceleration constraints need not be evaluated, and the overall algorithm is about as expensive as the application of a BDF method to the GGL-stabilized equations of motion. Moreover, the RATTLE method of molecular dynamics is included as special case. A convergence analysis of the presented α-RATTLE algorithm shows global second order in both position and velocity variables while the Lagrange multipliers are computed to first order accuracy. Additonally, the property of adjustable numerical dissipation carries over from the unconstrained case. [ABSTRACT FROM AUTHOR]

The article presents abstracts related to mathematics including the articles "Axisymmetric contact problems of the theory of elasticity for inhomogeneous layers," by Andrey Vasiliev and colleagues, "Shape sensitivity analysis of Timoshenko's plate with a crack under the nonpenetration condition," by N. P. Lazarev and E.M. Rudoy, and "A new direct time integration method for the equations of motion in structural dynamics," by John T. Katsikadelis.