Unified modal analysis of complex cable systems via extended dynamic stiffness method and enhanced computation

Free vibration of the complex cable system − An exact method using symbolic computation

Exact dynamic analysis of multi-segment cable systems

Two new methods for calculating vibration problems—the dynamic stiffness method and the mobility method—have been introduced in recent years. Each of these methods offers certain advantages over the older vibration theory. This article is an at tempt to unify these two methods so that the best features of each are included in one theory. By means of this theory it is possible to determine the resonance and antiresonance frequencies and the relative amplitudes of vibration of a mechanical system. The effect of the position of torque application on the vibration characteristics of a system are discussed. The application of the theory is demonstrated by a few examples including the problem of the pendulum damper.

Dynamic stiffness analysis of toroidal shells

The dynamic stiffness method is used to investigate the free vibration behaviour of a functionally graded beam (FGB). The material properties of the FGB are assumed to vary in the thickness direction based on a power-law. The kinetic and potential energies of the beam are formulated using the Timoshenko beam theory. The governing differential equations of motion in free vibration for the FGB are derived using Hamilton’s principle. The analytical expressions for axial force, shear force and bending moments at any cross-section of the FGB are obtained as a by-product of the Hamiltonian formulation. The differential equations are solved in closed analytical form for harmonic oscillation. The dynamic stiffness matrix of the FGB is then formulated by relating the amplitudes of forces and displacements at the ends of the beam. The Wittrick-Williams algorithm is used as solution technique to yield natural frequencies and mode shapes of the FGB. A parametric study is carried out by varying significant beam parameters and boundary conditions. The investigation required a substantial amount of validation exercise to confirm the predictable accuracy of the dynamic stiffness method. The results are discussed and some concluding remarks are made.

Harmonic response analysis of coupled plate structures using the dynamic stiffness method

The dynamic stiffness method is introduced to investigate the free vibration of laminated composite beams based on a third-order shear deformation theory which accounts for parabolic distribution of the transverse shear strain through the thickness of the beam. The exact dynamic stiffness matrix is found directly from the analytical solutions of the basic governing differential equations of motion. The Poisson effect, shear deformation, rotary inertia, in-plane deformation are considered in the analysis. Application of the derived dynamic stiffness matrix to several particular laminated beams is discussed. The influences of Poisson effect, material anisotropy, slenderness and end condition on the natural frequencies of the beams are investigated. The numerical results are compared with the existing solutions in literature whenever possible to demonstrate and validate the present method.

Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels

Modeling and simulation of aerial refueling by finite element method

<p>The Dynamic Finite Element (DFE) theory is employed to calculate the natural frequencies and mode shapes of three-layered sandwich beams. Three formulations are developed and tested using a number of different numerical cases. The first two theories pertain to the analysis of straight beams while the third one is developed for the analysis of curved beams. The results from all three derivations agree well with published data that uses Classical Finite Element (FEM) theory and the Dynamic Stiffness Method (DSM) in the analysis of the free vibration behaviour of sandwich beams. For all test cases, the DFE results agree well with the published results as well as the FEM developed as an additional benchmark for the DFE. The results of the research are very encouraging and demonstrate that DFE is a valuable structural analysis tool that can be used in conjunction with FEM and DSM. </p>

This study focuses on investigating the seismic performance of a reference 15 MW offshore wind turbine (OWT) supported by a large-diameter monopile, under various scour conditions. Fully coupled nonlinear finite element models considering 3D soil continuum for seismic response analysis of monopile-supported OWTs are very complicated and computationally expensive. Therefore, it is important to have good insight on the dynamic behaviour of monopile-supported OWTs subjected to earthquake excitations before performing high-fidelity fully nonlinear 3D simulations, which will not generally be practical in cases considering various seismic loading and scouring scenarios. The purpose of this study is to provide a fast and efficient tool to understand the interaction between the seismic behaviour of emerging large-capacity OWTs when scour erosion affects the dynamic properties of the structure. The dynamic stiffness method (DSM) is applied for the free vibration analysis of the structure. Modal superposition is used to calculate the seismic responses of the OWT by using the exact mode shapes obtained from the dynamic stiffness formulations. The rotor-nacelle-assembly (RNA) is idealized as a lumped mass at the tower top. The tower and monopile are modelled as Timoshenko beam-columns under axial compressive loading. The soil-structure interaction (SSI) is modelled using the Winkler framework, whereby the soil contribution is assumed as elastic to take advantage of the modal superposition method. Two different approaches are considered for the soil stiffness; a constant profile, and a linearly-varying subgrade modulus along the embedded depth of the pile, and both are appraised in this paper. Firstly, natural frequencies obtained from dynamic stiffness formulations are validated using experimental data from literature for the case of controlled dynamic SSI experiments, and for operating OWTs. Additionally, the natural frequency provided by a technical document for the reference 15 MW OWT is compared to the result from the DSM, where a very good agreement is observed. The seismic response time-histories of the OWT model are obtained under different input earthquake conditions, and subsequently verified by comparison with the results of finite element simulations. The effects of different scour scenarios and foundation modelling approaches on the free vibration and seismic responses are subsequently investigated. A parametric analysis is also performed using different soil properties to observe the interaction between the soil stiffness, scour condition, and seismic response. Finally, the influence of the frequency content of seismic records on the dynamic response of the OWT, specifically the tower top acceleration, the mudline rotation, and the mudline stress, is investigated considering various scour scenarios. The specific numerical case study undertaken for a reference 15 MW OWT, which is supported by a large monopile in very dense sand, shows that the seismic responses are very limited and that the elastic soil behaviour assumption is reasonable. Additionally, considering the computational efficiency, the proposed combination of DSM and modal superposition can be used for rapid seismic performance assessment of large diameter monopile-supported OWTs at the preliminary design stage to gain insight to the global dynamic response of the whole vibrating system.

This paper analyses free vibrations of framed nanostructures made of Functionally Graded Material (FGM) based on the Nonlocal Elastic Theory (NET) and the Dynamic Stiffness Method (DSM). FGM characteristics vary nonlinearly throughout the height of the beam element. The NET considers the nonlocal parameter that perfectly captured the size effect of nanostructures. However, the NET makes nonlocal paradoxes in the bending and vibration behaviour of framed nanostructures with the free ends. To overcome these phenomena, the nanostructure is modelled according to the Euler–Bernoulli beam theory and the variational-consistent nonlocal boundary conditions have been derived. The exact solutions of differential equations of motion and variational-consistent nonlocal boundary conditions are found using the DSM. The influences of the nonlocal, material, geometry parameters and Pasternak’s foundation on the free vibration are then analyzed. It is shown that the study can be applied to other FGMs as well as more complicated framed structures.

Dynamic resource allocation problems arise under a variety of settings. In “Survey of Dynamic Resource-Constrained Reward Collection Problems: Unified Model and Analysis,” Balseiro, Besbes, and Pizarro introduce a unifying model for a large class of dynamic optimization problems dubbed dynamic resource-constrained reward collection (DRC2) problems. Surveying the literature, they show that this class encompasses a variety of disparate and classical problems typically studied separately, such as dynamic pricing with capacity constraints, dynamic bidding with budgets, network revenue management, online matching, or order fulfillment. Furthermore, they establish that the DRC2 class is amenable to analysis by characterizing the performance of a central, certainty-equivalent heuristic. Notably, they provide a novel unifying analysis that isolates the drivers of performance, recovers as corollaries some existing specialized results, generalizes other existing results by weakening the assumptions required, and yields new results in specialized settings for which no such characterization was available.

It is meaningful to develop an accurate universal approach to solve the dynamical characteristic problem of a cable system combined with additional components used in real engineering. A segmented dynamic governing equation is presented for a taut shallow cable supported in the middle by a transverse component. Then, a universal frequency characteristic equation is proposed that considers all the influence factors, including the flexural rigidity, sagging, inclination angle, finite rigidity or damped boundary condition, and the intermediate supporting component. A general numerical solution is presented by giving an analytical derivative expression for the dynamic stiffness of the cable system. As a result, the modal parameters of an arbitrary complicated cable system can be obtained via the root of the equation. Finally, the accuracy of the proposed approach is verified using numerical cases, and its validity is also proved by applying it to case studies of several typical cable systems.

The use of the dynamic direct stiffness method (DDSM) for the harmonic analysis of beam structures is well known. In this paper the development of a harmonic plate element with similar characteristics is discussed. Whereas the finite-element method requires approximately six nodes per wavelength, the harmonic plate element requires nodes only at element connections, terminations, and load points. The governing equations are used in the variational form to derive the direct stiffness matrix. The matrix form of the element derived is compatible with those of the FEM and may be used for the harmonic analysis of general FEM/DDSM structures. The motivation for this development is the need for computationally efficient evaluation of the midfrequency range for general structures where the FEM method requires a large number of degrees of freedom and statistical energy analysis (SEA) does not provide sufficient accuracy. A number of examples are provided to establish verification of the DDSM plate element. Examples include the response of harmonically driven plates, plates with beam stiffeners, and plate/beam truss structures.