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In this study, the mechanical model of the circumferentially distributed rods and the nonlinear contact stiffness matrix which characterizes the contact effect between the disks are established. The contact stiffness matrix is composed of seven stiffness coefficients that characterize the lateral stiffness, shear stiffness, bending stiffness, and torsional stiffness of the contact interface. Then the influence of pre-tightening load, rod diameter, contact surface roughness, deformation phase and uneven pre-tightening load on stiffness coefficients is analyzed. Finally, combined with the experimentally verified nonlinear oil film force model based on short bearing theory, the dynamic model of the rod-fastening rotor-bearing system is established. By means of spectrum cascades and Campbell diagrams, the influence of the pre-tightening load, rod diameter, contact surface roughness and uneven pre-tightening load on the nonlinear dynamic characteristics of the rod-fastening rotor-bearing system is numerically analyzed. The results show that increasing the pre-tightening load can increase the contact stiffness, and make the dynamic characteristics of the rod-fastening rotor bearing system closer to that of the complete rotor bearing system. The increase in the diameters of rods significantly delay the partial separation of the contact surfaces. The smooth contact surface provides higher contact stiffness, but it is not conducive to maintaining the full contact state of the contact surfaces. The stiffness coefficients fluctuate with the deformation of the rotor, and the process is significantly affected by uneven pre-tightening load. Moreover, a self-excited vibration frequency f l accompanied by a combination frequency f b = 3 f l appears when the uneven pre-tightening load is caused by individual faulty rod. The number of rods, rod diameter, and contact surface roughness have similar effects on the dynamic characteristics of rod-fastening rotor bearing systems, which lead to fluent whirl and fluent whip appear in advance, reducing the stability of the system.

Significant machining errors may occur in thin-walled parts during the milling process due to their low rigidity. Conventionally, deformation prediction is conducted via the finite element method (FEM) due to its convenience and accuracy. However, it is extensively time-consuming because of the huge computation amount, which is caused by repeated computation of the displacements of all the nodes at each cutting location and iteration process. To solve this problem, a rapid deformation computation method based on stiffness matrix reduction was proposed. Furthermore, an iterative cutting force-induced error prediction model was established considering the tool-workpiece dynamic interaction, which shortened each cutting force-induced error prediction time from tens of seconds by traditional FEM to dozens of milliseconds. Lastly, the proposed method was applied to the compensation of thin-walled aluminum alloy blocks. The compensation experiment results revealed that the proposed model reduced the machining error by more than 53.6% with high real-time performance. This method demonstrates immense potential for further applications in deformation prediction of thin-walled freeform surface parts.

Broadband seismogram synthesis of the multi-layer and multi-scale has been a longstanding focus in seismic engineering area for decades, which also has many difficulties to overcome in computing range and efficiency. This paper proposes the direct exact stiffness matrix method, a form of the frequency-wavenumber method (abbreviated as fk ), for synthesizing seismograms. First, the direct exact stiffness matrix of the multi-layered and multi-scale underground structure is constructed in the frequency-wavenumber domain, and the sub-fault source (the earthquake fault is divided into the sub-fault source) is equivalent to the displacement-stress discontinuity to describe the dislocation. Furthermore, the “direct stiffness method” is used to determine the receiver’s motion, which establishes Green’s function (the relationship between the source and the receiver). The divergence index term in the stiffness matrix is modified to overcome the solution inefficiency associated with the thick layer and the high-frequency. The accuracy is verified by comparing the proposed method results with three examples in the publications, involving different source types and depth scales. The kinematics hybrid source model, based on the GP14.3 model, is introduced to generate a finite fault source model. The 2023 Turkey M w 7.8 earthquake source model is established, and the applicability of the fk method in the near-fault seismic simulation is validated by comparing it with the time history and spectrum records. The M w 7.0 dip-slip fault source model is based on the GP14.3 hybrid source generation which is set in a multi-scale layered half-space. The study area comprises a near-fault area with 122 receivers. Conclusions are drawn based on the analysis of the normalized response spectrum, peak spectral acceleration, and difference rate by changing the shallow velocity structure. The results show that the shallow velocity structure results in an average amplification of 5.1% in the fault-parallel direction and 7.1% in the fault-perpendicular direction.

We present a set of advanced analytical formulations that facilitates the accurate analysis and efficient implementation of finite deformable thin Kirchhoff–Love beams. This paper enhances the prevailing differential geometry based large deformation beam models by producing geometrically exact formulations for initial curvatures, non-zero force tangents and external stiffness matrix contributions of spatial beams. Though it is not analytically merged in existing beam models, initial curvatures of beams have a significant influence on the integration of forces over beam cross-sections. We reveal this influence through the systematic deduction of the Jacobian in volume integrals of beam forces. Also, this paper demonstrates the applicability of follower loads on beams with necessary adjustments to the global Hessian matrix. We adopt the isogeometric analysis formalism in beam body discretisation and algorithmic implementation of the presented formulations.

This study presents a new coupling model (CM) of finite element (FE) and peridynamics (PD), in which only very few PD nodes exist. In the coupling model, PD subregion is directly coupled with FE subregion without an overlapped zone, and the force is transferred between PD nodes and finite elements by a connection stiffness matrix. Since dynamic transformation technique is implemented, PD subregion is adaptively generated and evolved, and ensure that the whole damage process is completed. In addition, as an optimization of the coupling model, a densified-material-point model (DMPM) is achieved, which can remove the limitation of element type and enhance the flexibility of the coupling model. As a result, the computational efficiency of coupling algorithms will be greatly improved, and numerical error can be overcome in inferring the damage region. The capability of the developed coupling model was demonstrated by the stretch examples of plates with different discrete cases, and damage analysis was further conducted to demonstrate the strong capability of the DMPM in capturing failure mode.

PurposeIn this paper, the main factors influencing the structure stiffness will be analyzed by studying the tangent stiffness matrix based on different requirement in engineering practice. The authors can obtain the deformation of three-bar tensegrity basic unit in different load, and gain the primary factor by comparing the deformation, which will provide reference to concrete structure design in the engineering.Design/methodology/approachThe mathematical model of tensegrity structure was built by establishing generalized node coordinates and connective matrix. Three main factors that affect the structure deformation can be obtained by analyzing the stiffness matrix, which is preload, Young's modulus, and cross-sectional area, the thinking of deformation also be sorted out. The deformation analysis of the concrete structure is carried out, and it is concluded that increasing the cross-sectional area can quickly improve the stiffness of the structure, which provides a reference for the structural variable stiffness design in practical engineering.Findings(1) When the axial external force is applied to the structure, the torsion-angle deformation of the structure is the largest, and the radial deformation of the structure is the smallest. (2) The structure stiffness can be rapidly enhanced by increasing the cross-sectional area. But the cross-sectional area can't be increased indefinitely. Because the mass will be increased once increasing the cross-sectional area, which will destroy the structure of the advantages of light weight in engineering practice.Originality/valueThe deformation analysis of the concrete structure is carried out, and it is concluded that increasing the cross-sectional area can quickly improve the stiffness of the structure, which provides a reference for the structural variable stiffness design in practical engineering.

In finite element analysis (FEA), optimizing the storage requirements of the global stiffness matrix and enhancing the computational efficiency of solving finite element equations are pivotal objectives. To address these goals, we present a novel method for compressing the storage of the global stiffness matrix, aimed at minimizing memory consumption and enhancing FEA efficiency. This method leverages the block symmetry of the global stiffness matrix, hence named the blocked symmetric compressed sparse column (BSCSC) method. We also detail the implementation scheme of the BSCSC method and the corresponding finite element equation solution method. This approach optimizes only the global stiffness matrix index, thereby reducing memory requirements without compromising FEA computational accuracy. We then demonstrate the efficiency and memory savings of the BSCSC method in FEA using 2D and 3D cantilever beams as examples. In addition, we employ the BSCSC method to an engine connecting rod model to showcase its superiority in solving complex engineering models. Furthermore, we extend the BSCSC method to isogeometric analysis and validate its scalability through two examples, achieving up to 66.13% memory reduction and up to 72.06% decrease in total computation time compared to the traditional compressed sparse column method.

In this paper, we compute stiffness matrix of the nonlocal Laplacian discretized by the piecewise linear finite element on nonuniform meshes, and implement the FEM in the Fourier transform domain. We derive useful integral expressions of the entries that allow us to explicitly or semi-analytically evaluate the entries for various interaction kernels. Moreover, the limiting cases of the nonlocal stiffness matrix when the interactive radius δ→0 or δ→∞ automatically lead to integer or fractional FEM stiffness matrices, respectively, and the FEM discretisation is intrinsically compatible. We conduct ample numerical experiments to study and predict some of its properties and test on different types of nonlocal problems. To the best of our knowledge, such a semi-analytic approach has not been explored in literature even in the one-dimensional case.

Using variable stiffness design to enhance the bearing capacity of composite structures has been an appealing strategy. In previous approaches for optimizing discrete fiber orientation, a penalty strategy is always adopted to drive the design variables to 0 or 1, which is unstable and does not guarantee the convergence of all fiber orientations. This paper proposes a novel optimization method termed sequential gradient chase (SGC) for optimizing discrete fiber orientations in composites based on stiffness matrix interpolation. We construct an optimization domain tightening criterion and a discrete direction search criterion. Constraints are incorporated into the solver by using an intriguing update rule for design variables, resulting in conversion from a constrained optimization problem to an unconstrained optimization problem. The suggested method is capable of achieving convergence of all fiber orientations without the use of any penalty measure and producing a clear fiber layout scheme devoid of gray-scale elements. Numerical examples demonstrate that the proposed method can be successfully applied to a variety of physical models for the calculation of in-plane stress and strain, plate bending, and shell torsion, and possesses a stable optimization ability and a high solving efficiency.

Micro-electro-mechanical-systems (MEMS) extensively employed planar mechanisms with elastic curved beams. However, using a curved circular beam as a flexure hinge, in most cases, needs a more sophisticated kinetostatic model than the conventional planar flexures. An elastic curved beam generally allows its outer sections to experience full plane mobility with three degrees of freedom, making complex non-linear models necessary to predict their behavior. This paper describes the direct kinetostatic analysis of a planar gripper with an elastic curved beam is described and then solved by calculating the tangent stiffness matrix in closed form. Two simplified models and different contributions to derive their tangent stiffness matrices are considered. Then, the Newton–Raphson iterative method solves the non-linear direct kinetostatic problem. The technique, which appears particularly useful for real-time applications, is finally applied to a case study consisting of a four-bar linkage gripper with elastic curved beam joints that can be used in real-time grasping operations at the microscale.