Equivalent Stiffness Matrices Research Articles

Integration of 3-D 4-directional braided composites into dynamic modeling of plates with large overall motions

In this study, a dynamic model of a 3-D 4-directional braided composite plate with large overall motions is proposed. Based on the Kirchhoff’s plate hypothesis, the deformation of the orthotropic plate is described. The assumed mode method is employed to discretize the displacement variables, and the rigid-flexible coupled dynamic model of the plate is derived from the Lagrange equation. The equivalent stiffness matrices of 3-D 4-directional braided composites with different braided angles are obtained using the homogenization method. The effects of different velocities and braided angles on the dynamic characteristics of the plate with translational motion and fixed-axis rotation, respectively, are investigated. The present study lays the theoretical foundation for analyzing the dynamic properties of orthotropic plates in engineering components. For example, the dynamic model established in this study can be used to estimate the dynamic responses of braided composite parts in aeronautical structures.

Extended multiscale FEM-based concurrent optimization of three-dimensional graded lattice structures with multiple microstructure configurations

In this work, a novel concurrent design method for three-dimensional graded lattice structures considering multiple microstructure configurations is proposed. Firstly, the Multiple Variable Cutting (M-VCUT) level set method is employed to represent the geometry of lattice microstructures and define the optimization parameters. Then, the extended multiscale finite element method (EMsFEM) is implemented for calculating the equivalent stiffness matrices of microstructures in offline form. Compared with the traditional homogenization-based multiscale approaches, the proposed method eliminates the scale separation assumptions and takes into account the coupling effects in different directions of materials. In addition, a data-driven method and a parallel computing approach are integrated together so that computational efficiency can be greatly improved. After the iteration process, the optimized full-scale graded structures are reconstructed using interpolation technology to generate well-connected microstructures in neighboring cells. Finally, several 3D numerical examples are utilized to validate the effectiveness of the proposed method, where multiple lattice microstructure configurations are considered, including truss-lattices and plate-lattices.

High fidelity FEM based on deep learning for arbitrary composite material structure

Due to the outstanding performance, composite materials are widely used and analyzing their properties and designing them based on performance has become a crucial task in the field of many manufacturing industries. Composite materials possess complex multiscale structures, and traditional fine-scale finite element modeling and analysis may lead to severe computational resource challenges. To overcome this difficulty, breakthroughs in key technologies of multiscale accelerated analysis algorithms are required. In this study, an innovative approach based on artificial intelligence and multiscale finite element method is presented. This approach involves partitioning the entire composite material structure into coarse grids that resemble homogenous structures of similar size, providing results consistent to fine-grid finite element analysis. By utilizing CNN for image feature recognition and employing the CGAN adversarial method, coarse-grid equivalent stiffness matrices and multiscale shape functions from completely random microstructures of composite materials can be obtained. Consequently, this enables a rapid response process from microstructure to low-resolution grid to high-resolution physical field, with remarkably accurate physical field results. Moreover, compared to traditional fine-grid finite element methods, this approach significantly reduces memory usage and computation time. This method is applicable to composite materials with varying shaped inclusions, different component properties, and diverse geometric distributions, allowing these materials to perform high-fidelity finite element calculations on coarse grids and predict their mechanical behavior. Furthermore, this breakthrough opens avenues for accelerating the optimization design of composite materials with diverse mechanical functionalities, by employing a bottom-up approach.

A coupled SBFETI-BDEs method for solving 2D static and dynamic contact problem with friction

In this study, we propose a coupled approach called the SBFETI-BDEs method for solving the 2D elastic frictional contact problem. The contact system is decomposed into multiple subdomains, similar to the finite-element tearing and interconnecting (FETI) algorithm. Each subdomain can be regarded as a scaled boundary finite element (SBFE), where only the boundaries are discretized. To avoid computing the global contact flexibility matrix using the force method, we consider the generalized inverse of the stiffness matrix for each subdomain as a subcontact-flexibility matrix. We uniformly formulate the contact equations, displacement continuity equations, and the equilibrium equations of force for the floating subdomains, as a B-differentiable equations (BDEs) system, which we solve using the B-differentiable damped Newton-Raphson method that possesses a global convergence property. The proposed method offers increased computational efficiency without compromising accuracy when compared to the original BDEs algorithm without domain decomposition. We present the dynamic scheme of the SBFETI-BDEs method within the framework of the HHT time-integration scheme, where the equilibrium equations of force for the floating subdomains disappear due to their nonsingular equivalent stiffness matrices. Numerical examples are provided to demonstrate the convergence, accuracy, efficiency, and capability in solving engineering problems of the proposed method.

Dynamic modeling of a 3-D 4-directional braided composite beam with a central rigid body

Although 3-D braided composites can be incorporated with rotating central body-beam systems, there are currently few dynamic models for the rotating systems made of braided composites. This paper proposes a dynamic model of a 3-D 4-directional braided composite beam with a central rigid body. Based on the Timoshenko beam hypothesis, the displacements of the composite beam are described by the arc-length coordinate, and the displacement field is represented by the slope angle, tensile strain, and shear angle. The assumed mode method is employed to discrete the displacement field, and the dynamic model of the system is derived from the Lagrange equation. The equivalent stiffness matrices of 3-D 4-directional braided composite beams with different braiding angles are obtained using the volume average method. The influences of different angular velocities, braiding angles, and beam section parameters on the system's dynamic characteristics are studied. The model proposed in this study can be utilized to estimate the dynamic responses of braided composite blades, and lays the theoretical foundation for analyzing dynamic characteristics of anisotropic beams in engineering.

Structural Analysis of the US ITER ECH Transmission Line System

The electron cyclotron heating (ECH) and current drive system is one of the main plasma heating systems for ITER. It uses high-power microwave beams with the power deposition location steerable across the plasma cross section. Microwave power is conveyed via transmission lines (TLs) that run from the gyrotrons in the radio frequency building through the assembly hall and tokamak building to the ECH launchers within the tokamak vacuum vessel. The ECH system includes a vast array of interconnected TL waveguides, in-line components, and support structures. Finite element (FE) modeling provides an essential means of simulating the system, applying loads and determining deflections, rotations, forces, moments, and stresses in order to evaluate various structural and microwave transmission performance metrics. A representative FE model of the overall ECH TL system is developed in ANSYS®. This top-level model defines the centerline of the waveguide system. Waveguide segments are represented by line elements (beams and pipes) with equivalent section properties, and support structures are represented by boundary conditions. A systematic approach is used to model each ECH component with lumped masses and structurally equivalent stiffness matrices or ANSYS superelements. The top-level TL FE model is used to evaluate the various loads (thermal, vacuum, seismic, etc.) and operating scenarios. The top-level model directly calculates stresses in the straight aluminum waveguide segments. The model provides the forces and moments acting on the in-line components for detailed submodel assessments. Displacement results from the top-level analysis feed into a separate microwave performance model to help determine operational efficiency. All TL performance and thermal-structural requirements are met, as specified by the applicable codes and standards, and successfully documented in numerous technical reports and demonstrated at the final design review.

Extended continuum model for dynamic analysis of beam-like truss structures with geometrical nonlinearity

An approach to extend the application of the continuum model to beam-like lattice truss structures with geometrical nonlinearity is presented in this paper. The geometrical nonlinearity within the truss structures is accounted for by dividing it into substructures and introducing a co-rotational coordinate system to decompose the substructure motion into rigid body motion and pure deformations. The linear continuum model of each substructure is derived using the continuum model based on the small deformation hypothesis in the local coordinate system, in which the deformation of the truss structure is still small. Then, the linear mass and stiffness matrices of the continuum elements are converted into the nonlinear equivalent mass and stiffness matrices for the global coordinate system. The equations of motion for the truss structure are established using Lagrange's equations. Three numerical simulation examples are considered and the results show that the proposed model requires fewer degrees of freedom and has better efficiency than the finite element model based on the truss elements while also obtaining the desired accuracy.

In-Plane and Out-of-Plane Elastodynamics of Thin Rings and Seals

Thin curved rings used mostly as seals, including in internal combustion engines undergo complex elastodynamic behavior when subjected to a combination of normal radial loading and tangential shear with friction. In turn, their complex modal behavior often results in loss of sealing, increased friction, and power loss. This paper presents a new finite difference approach to determine the response of thin incomplete circular rings. Two interchangeable approaches are presented; one embedding mass and stiffness components in a unified frequency-dependent matrix, and the other making use of equivalent mass and stiffness matrices for the ring structure. The versatility of the developed finite difference formulation can also allow for efficient modification to account for multiple dynamically changing ring support locations around its structure. Very good agreement is observed between the numerical predictions and experimental measurements, particularly with new precision noncontact measurements using laser Doppler vibrometry. The influence of geometric parameters on the frequency response of a high performance motorsport engine's piston compression ring demonstrates the degree of importance of various geometrical parameters on ring dynamic response.

Topology optimization of plate/shell structures with respect to eigenfrequencies using a biologically inspired algorithm

ABSTRACTIn this article, a novel approach is presented to perform topology optimization in a simple and explicit way. The method capitalizes on the use of a bio-inspired algorithm to represent topology, leading to more flexible optimization solutions along with explicit structure representation. To avoid remeshing upon design changes, a special treatment called the enhanced stiffness transformation approach (ESTA) is introduced to transform the stiffness and mass matrices of the growing stiffener into a set of equivalent stiffness and mass matrices. In this way, stiffeners are separated from the finite element mesh and can grow in an arbitrary direction to form an optimized layout solution. Notably, this approach incorporates more geometric information into topology optimization, which improves the clarity of stiffener layouts. Finally, the effectiveness of the proposed method is illustrated with two examples of maximum eigenfrequency design of plate/shell structures.

Numerical simulation of drive axles considering the nonlinear couplings between gears and bearings

The drive axle is the core of the powertrain in automotive vehicles, and numerical simulation is a common and effective approach for designing and analysing drive axles. This article develops a new numerical simulation method for drive axles considering the nonlinear couplings between gears and bearings. Two types of models are proposed and integrated: a reduced model with nonlinear bearing elements and equivalent gear elements, which is efficient for nonlinear bearing simulation, and a gear contact analysis model with equivalent bearing stiffnesses, which can accurately reflect the gear mesh conditions and system deflection. The equivalent gear mesh parameters used in the reduced model are derived from the contact analysis model, and the linear equivalent bearing stiffness matrices used in the contact analysis model are derived from the reduced model. Therefore, the static equilibrium condition of the entire drive axle system is obtained through an iterative calculation process involving the two models under the same load condition. The good correlation between the numerical and experimental results of the gear-loaded contact pattern indicates that the proposed method is reliable.

An Inhomogeneous Cell‐Based Smoothed Finite Element Method for Free Vibration Calculation of Functionally Graded Magnetoelectroelastic Structures

To overcome the overstiffness and imprecise magnetoelectroelastic coupling effects of finite element method (FEM), we present an inhomogeneous cell‐based smoothed FEM (ICS‐FEM) of functionally graded magnetoelectroelastic (FGMEE) structures. Then the ICS‐FEM formulations for free vibration calculation of FGMEE structures were deduced. In FGMEE structures, the true parameters at the Gaussian integration point were adopted directly to replace the homogenization in an element. The ICS‐FEM provides a continuous system with a close‐to‐exact stiffness, which could be automatically and more easily generated for complicated domains, thus significantly decreasing the numerical error. To verify the accuracy and trustworthiness of ICS‐FEM, we investigated several numerical examples and found that ICS‐FEM simulated more accurately than the standard FEM. Also the effects of various equivalent stiffness matrices and the gradient function on the inherent frequency of FGMEE beams were studied.

New ways to symmetrise free vibration equations of fluid–structure interaction systems

It can be difficult to use finite-element (FE) analyses to solve fluid–structure interaction due to asymmetric equivalent mass and stiffness matrices. Based on mathematical calculations, two new techniques are proposed to achieve a symmetric eigenvalue problem. For this purpose, new scale matrices to symmetrise the interaction system's eigenproblem are presented. Numerical FE analysis of a liquid-storage tank proves the validity and efficiency of the proposed techniques compared with other existing approaches, providing higher convergence rate and lower analysis time.

Change mechanism of solver performance: With a special focus on time history analysis of supertall buildings

SummaryThe correlation between the performance of some commonly used linear system solvers and the characteristics of system matrices, especially in the time history analysis of supertall buildings, is comprehensively identified. Relying on 5 well‐designed structural models with height varying from 47 to 660 m, the supernodal LLT solver and the conjugate gradient (CG) solver with the factorized sparse approximate inverse preconditioner are selected due to their actual efficiency. More importantly, basing on 90 fictitious supertall structural models generated from the parameterized method, the effects of structural characteristics on solver performance and the corresponding mechanism are studied by 3 mathematical indexes with parameters come from the equivalent dynamic stiffness matrices. Impacts of processors on the solvers are also clarified by associating the architectural characteristics of processors with the indexes. The performance of the 2 solvers is affected globally with increasing building size; different variation laws are shown for different structural systems. However, the accuracy of the CG solver does not exhibit distinct variation due to a slight change in the 2‐norm condition number of the matrices. The CG solver is found to be more applicable on GPU processor than the LLT solver due to its smaller gap between computational workload and the amount of data transferred.

The stiffness spreading method for layout optimization of truss structures

A novel optimization method, stiffness spreading method (SSM), is proposed for layout optimization of truss structures. In this method, stiffness matrices of the bar elements in a truss structure are represented by a set of equivalent stiffness matrices which are embedded in a weak background mesh. When the proposed method is used, it is unnecessary for the bar elements in a truss structure to be connected to each other during the optimization process, and each of the bar elements can move independently in the design domain to form an optimized design. Another feature of the method is that the sensitivity analysis can be done analytically, making gradient based optimization algorithms applicable in the solution. This method realizes the size, shape and topology design optimization of truss structures simultaneously and allows for more flexibility in topology change. Numerical examples illustrate the feasibility and effectiveness of the proposed method.

An Approximate Method for Permuting Frame with Repeated Lattice Structure to Equivalent Beam

In this paper, equivalent stiffness and mass matrices of a beam to a frame with repeated lattice structure are obtained to permute a frame to an equivalent continuous beam. Static and dynamic results of the permutated beam are obtained and compared to ANSYS's results for the original frames with good agreements. The technique can be usefully employed for analyzing the global behavior of a frame by permuting to a beam. The technique also leads to reducing significantly the degrees of freedom of frames and thus the computing time in their dynamic analysis.

Accurate identification of the frequency response functions for the rotor–bearing–foundation system using the modified pseudo mode shape method

In this paper, an identification technique in the dynamic analyses of rotor–bearing–foundation systems called the pseudo mode shape method (PMSM) was improved in order to enhance the accuracy of the identified dynamic characteristic matrices of its foundation models. Two procedures, namely, phase modification and numerical optimisation, were proposed in the algorithm of PMSM to effectively improve its accuracy. Generally, it is always necessary to build the whole foundation model in studying the dynamics of a rotor system through the finite element analysis method. This is either unfeasible or impractical when the foundation is too complicated. Instead, the PMSM uses the frequency response function (FRF) data of joint positions between the rotor and the foundation to establish the equivalent mass, damping, and stiffness matrices of the foundation without having to build the physical model. However, the accuracy of the obtained system's FRF is still unsatisfactory, especially at those higher modes. In order to demonstrate the effectiveness of the presented methods, a solid foundation was solved for its FRF by using both the original and modified PMSM, as well as the finite element (ANSYS) model for comparisons. The results showed that the accuracy of the obtained FRF was improved remarkably with the modified PMSM based on the results of the ANSYS. In addition, an induction motor resembling a rotor–bearing–foundation system, with its housing treated as the foundation, was taken as an example to verify the algorithm experimentally. The FRF curves at the bearing supports of the rotor (armature) were obtained through modal testing to estimate the above-mentioned equivalent matrices of the housing. The FRF of the housing, which was calculated from the equivalent matrices with the modified PMSM, showed satisfactory consistency with that from the modal testing.

Efficient computation for dynamic responses of systems with time-varying characteristics

Based on Neumman series and epsilon-algorithm, an efficient computation for dynamic responses of systems with arbitrary time-varying characteristics is investigated. Avoiding the calculation for the inverses of the equivalent stiffness matrices in each time step, the computation effort of the proposed method is reduced compared with the full analysis of Newmark method. The validity and applications of the proposed method are illustrated by a 4-DOF spring-mass system with periodical time-varying stiffness properties and a truss structure with arbitrary time-varying lumped mass. It shows that good approximate results can be obtained by the proposed method compared with the responses obtained by the full analysis of Newmark method.

Combined Torsional-Bending-Axial Dynamics of a Twisted Rotating Cantilever Timoshenko Beam With Contact-Impact Loads at the Free End

In this paper, consideration is given to the dynamic response of a rotating cantilever twisted and inclined airfoil blade subjected to contact loads at the free end. Starting with the basic geometrical relations and energy formulation for a rotating Timoshenko beam constrained at the hub in a centrifugal force field, a system of coupled partial differential equations are derived for the combined axial, lateral and twisting motions which includes the transverse shear, rotary inertia, and Coriolis effects, as well. In the mathematical formulation, the torsion of the thin airfoil also considers a very general case of shear center not being coincident with the CG (center of gravity) of the cross section, which allows the equations to be used also for analyzing eccentric tip-rub loading of the blade. Equations are presented in terms of axial load along the longitudinal direction of the beam which enables us to solve the dynamic pulse buckling due to the tip being loaded in the longitudinal as well as transverse directions of the beam column. The Rayleigh–Ritz method is used to convert the set of four coupled-partial differential equations into equivalent classical mass, stiffness, damping, and gyroscopic matrices. Natural frequencies are computed for beams with varying “slenderness ratio” and “aspect ratio” as well as “twist angles.” Dynamical equations account for the full coupling effect of the transverse flexural motion of the beam with the torsional and axial motions due to pretwist in the airfoil. Some transient dynamic responses of a rotating beam repeatedly rubbing against the outer casing is shown for a typical airfoil with and without a pretwist.

Evaluating foundation mass, damping and stiffness by the least‐squares method

AbstractThis paper discusses how to use the three‐dimensional (3D) time‐domain finite‐element method incorporating the least‐squares method to calculate the equivalent foundation mass, damping and stiffness matrices. Numerical simulations indicate that the accuracy of these equivalent matrices is acceptable when the applied harmonic force of 1+sine is used. Moreover, the accuracy of the least‐squares method using the 1+sine force is not sensitive to the first time step for inclusion of data. Since the finite‐element method can model problems flexibly, the equivalent mass, damping and stiffness matrices of very complicated soil profiles and foundations can be established without difficulty using this least‐squares method. Copyright © 2003 John Wiley & Sons, Ltd.

Small Vibrations of Flexible Bars by Using the Finite Element Method with Equivalent Uniform Stiffness and Mass Methodology

The object of this research is to calculate natural frequencies and mode shapes of a small oscillation vibration, superimposed on large static displacement, by using the finite element method in conjunction with the equivalent uniform stiffness and mass method. The equivalent uniform stiffness and mass matrices are determined from the differential equation by using Galerkin's method. The analysis herein is three-fold. Firstly, the large static equilibrium configuration of the deformed flexible beam is established. This is achieved by using the Euler-Bernoulli equation in conjunction with equivalent pseudo-linear and/or equivalent non-linear systems of constant stiffness. This concept and methodology were used quite extensively by the authors to study large deformation characteristics of various flexible beam problems of uniform and variable stiffness and of arbitrary loading conditions (see, for example, the work of Fertis and Afonta [1, 2], and Fertis and Lee [3, 4]). Secondly, the differential equation of motion for small amplitude vibrations from the large static equilibrium configuration is derived. Galerkin's technique is used to approximate the differential equation of motion. Finally, the natural frequencies of vibration and the associated mode shapes are determined by solving the resulting eigenvalue problem. A canned eigensolver from the University of Akron IMSL is used to extract the eigenvalues and eigenvectors. The concept of equivalent pseudo-linear systems of constant stiffness is used extensively in this research, in order to simplify the solution of these complex non-linear problems.