System Stiffness Matrix Research Articles

Natural frequency analysis of gear system with interval shaft misalignment in multidimensions

Due to the existing tolerances, the shaft misalignment of gear system is inevitably uncertain and shows interval property, which will cause the natural frequency to fluctuate in an interval range, therefore the traditional dynamics analysis that takes it as a deterministic parameter cannot reflect the interval characteristics of natural frequency. To overcome this conundrum, an improved interval eigenvalue algorithm based on matrix non-negative factorization and the Chebyshev inclusion function was proposed to find the natural frequency range of gear system under such uncertainty. First, the average meshing stiffness range of the gears with interval shaft misalignment is obtained by the Chebyshev interval approach. Then combined with the consideration of the remaining excitation parameter deviation coefficients, the global interval stiffness matrix and mass matrix of a single-stage gear system are obtained. Afterwards, the non-negative factorizations of these two interval matrices are carried out and parameter vertex theory is finally employed to obtain the upper and lower bounds of the eigenvalues. The feasibility of the applied algorithm is verified through comparison with the scanning method. Numerical results show that the uncertainty of shaft misalignment will enormously widen the system's resonance bandwidth and intensify oscillation, these will provide a new and valuable perspective for the tolerance allocation in gearbox design.

Atomistically informed hierarchical modeling for revisiting the constituent structures from heredity and nano-micro mechanics of sheath-core carbon fiber.

To better understand the heterogeneous anisotropic nanocomposite features and provide reliable underlying constitutive parameters of carbon fiber for continuum-level simulations, hierarchical modeling approaches combining quantum chemistry, molecular dynamics, numerical and analytical micromechanics are employed for studying the structure-performance relationships of the precursor-inherited sheath-core carbon fiber layers. A robust debonding force field is derived from energy matching protocols, including bond dissociation enthalpy calculations and rigid-constraint potential energy surface scan. Logistic long range bond stretching curves with exponential parameters and shifted force vdW curves are designed to diminish energy perturbations. The pseudo-crystalline microstructure is proposed and validated using virtual wide angle X-ray diffraction patterns and bond-orientational order parameters. The distribution or alignment features of the nanocomposite microstructures are collected from quantum chemical topology analysis and normal vector extractions. Non-equilibrium tensile loading simulation predicts the decomposed strain energy contributions, principal-axis modulus, strength limit, localized stress, and fracture morphologies of the model. Finally, an atomistically-informed stiffness prediction model combining numerical homogenization and analytical self-consistent Eshelby-Mori-Tanaka-type effective mean field micromechanics theory is proposed, giving a successful estimation of the overall stiffness matrix of the sheath-core carbon fiber system. The hierarchical models in combination with the carbonization reaction template will help in providing efficient and feasible schemes for the synergistic process-performance control of distinct types of carbon fiber.

Uncoupling of Linear Multi-Degree-of-Freedom Damped Gyroscopic Potential Systems

Abstract This paper deals with the uncoupling of linear damped multi-degree-of-freedom gyroscopic potential systems in which the damping is taken to have a specifically chosen form. Necessary and sufficient conditions are obtained that guarantee the uncoupling of such damped systems into independent subsystems with at most two degrees-of-freedom. Along with several other results, it is shown that when the potential (stiffness) matrix of the damped system has distinct eigenvalues—a situation commonly found in civil, mechanical, and aerospace engineering, as well as in nature—the damping matrix must have this specifically chosen form for any such multi-degree-of-freedom system to be capable of being uncoupled.

AN EFFICIENT METHOD TO SIMULATE THE VARYING STATIC RECEPTANCE OF A THIN-WALLED WORKPIECE

A new method to simulate the varying static receptance of a thin-walled workpiece based on the Cholesky decomposition is presented in this paper. In this method, the system stiffness matrix is updated by subtracting the elemental stiffness matrices of the removed elements from the initial system stiffness matrix. The updated system stiffness matrix is permuted by using a novel method based on element removing sequence and considering fill-in reduction. The Cholesky factor of the permuted system stiffness matrix is reused to compute the static receptance of the workpiece for multiple cutter locations. The reduction of computation time by using this new method is proved by a numerical test and the accuracy of the simulation is demonstrated with a machining test.

Configuration Optimization and Distributed Formation Control for Tethered Multirotor UAS

This article improves the robustness of tethered multirotor unmanned aircraft systems (UASs) through configuration optimization and controller design. Capacity margin and stiffness matrix are defined to measure the robustness of the system. The relationship between capacity margin and system configuration is analyzed based on a new optimization method. Moreover, the mathematical expression of system stiffness matrix argument with configuration is derived. Based on these two relations, the optimal configuration is determined to strengthen the robustness of the system. Under composite disturbances, it is not trivial to maintain the designed optimal configuration. Thus, a disturbance observer (DO) is developed to eliminate the disturbance generated by the tension force, which is a constant or harmonic signal. In addition, a fixed-time extended state observer (FTESO) is designed to mitigate the external aerodynamic force acting on the UAS. Accordingly, a distributed formation control algorithm based on the designed DO and FTESO is constructed for formation tracking control. Finally, the indoor experiments are demonstrated to validate the proposed strategy.

Component Mode Synthesis Based on the Energy Method for Frequency Domain Dynamic Response Analysis of Ballastless Track Structure

Used extensively in structural dynamics analysis, the energy method is advantageous for transforming boundary value problems of differential equations into variational extremum problems. Recent applications include analyzing the wave and vibration characteristics of track structures. However, traditional energy methods for dynamic modeling and analysis of ballastless track structures require obtaining the global stiffness matrix and mass matrix of the entire coupled system, which decreases computational efficiency. The EM-CMS algorithm for frequency domain dynamic response analysis of ballastless track structures has been proposed to address this issue. The core of EM-CMS is the development of model reduction strategies within the framework of the energy method to reduce matrix dimensions and thereby improve computational efficiency. Specifically, the steel spring floating slab track is the focus of this research. Utilizing the energy functional variational method, the modal properties of the rail and floating slab structures are obtained, and truncation is performed. A reduced-order model for the steel spring floating slab track is established by considering the boundary conditions between the rail-floating slab and the floating slab foundation (connected through fastener springs and steel springs, respectively, and considering the springs’ elastic potential energy). Comparatively, the computational efficiency of EM-CMS is approximately ten times greater than that of the energy method. The method’s accuracy is also carefully validated against finite element simulations.

РАСЧЕТ ОСАДОК ПРОТЯЖЕННОГО ПЛИТНОГО ФУНДАМЕНТА КОНЕЧНОЙ ЖЕСТКОСТИ НА ОСНОВЕ ДАННЫХ КОМПЬЮТЕРНОГО МОДЕЛИРОВАНИЯ

The paper presents a database for estimation the magnitude of the settlement of an extended slab foundation of finite stiffness in the initial phase of projecting. The offered solution uses only interpolation methods without involving powerful computing devices. The computer program FEA, which formalises the finite element method, was used to obtain the result. The maximum horizontal and vertical dimensions of the computational schemes were 25H3•46H3. The computational scheme consisted of 57 500 finite elements in the form of right isosceles triangles conjugated at 29 106 nodes. The width of the stiffness matrix of the linear equation system was 466. This made it possible to eliminate the influence of trivial boundary conditions on the results. 560 computational operations were performed. It corresponded to the number of possible combinations of numerical values of the variables calculated parameters adopted in computer modelling of process of the slab foundation settlement. As an upshot of the computations, the authors made a table of coefficients of approximating expressions for the curves constructed from the results of the calculated dependences of slab foundation settlement on the variable design parameters. The article provides a calculation of settlements of two extended slab foundations of different widths, using a constructed database and interpolation methods. The control calculation in the FEA showed the difference between the values of the control settlements and the values obtained using the proposed tables and the linear interpolation method for 13.45 and 22.08 %. Summing up the results, the offered database can be used for preliminary (estimation) calculations of extended slab foundations settlements.

System reduction-based approximate reanalysis method for statically indeterminate structures with high-rank modification

Efficient structural reanalysis for high-rank modification plays an important role in engineering computations which require repeated evaluations of structural responses, such as structural optimization and probabilistic analysis. To improve the efficiency of engineering computations, a novel approximate static reanalysis method based on system reduction and iterative solution is proposed for statically indeterminate structures with high-rank modification. In this approach, a statically indeterminate structure is divided into the basis system and the additional components. Subsequently, the structural equilibrium equations are rewritten as an equation system with the stiffness matrix of the basis system and the pseudo forces derived from the additional elements. With the introduction of the spectral decomposition, a reduced equation system with the element forces of the additional elements as unknowns is established. The approximate solutions of the modified structure can then be obtained by solving the reduced equation system through a pre-conditioned iterative solution algorithm. The computational costs of the proposed method are compared with those of two other reanalysis methods, and numerical examples including static reanalysis and static nonlinear analysis are presented. The results demonstrate that the proposed method has excellent computational performance for both structures with homogeneous material and structures composed of functionally graded beams. Furthermore, the superiority of the proposed method suggests that the combination of system reduction and pre-conditioned iterative solution technology is an effective approach to develop high-performance reanalysis methods.

A new, fast method for solving finite-element equations iteratively based on Gauss–Seidel

Solving large equation systems is the most time-consuming part of finite-element modelling – iterative techniques are favoured for models with numerous degrees of freedom where direct techniques have high storage requirements. Classical iterative techniques such as Gauss–Seidel (GS) are robust due to guaranteed convergence and algorithmic simplicity. Performance of iterative techniques chiefly depends on system scale and stiffness matrix properties – which are influenced by structural configuration. However, it is possible to adjust an iterative algorithm such that its speed is greatly enhanced for a certain class of structural configurations. This paper presents an adjusted GS solver, ‘constrained Gauss–Seidel’ (CGS), which has been formulated to solve typical multi-storey structures with an enhanced speed. The innovation in CGS originates from the adoption of a diaphragmatic relaxation mechanism that results in dividing equations into two groups to optimally deal with the different unknown types. In this paper, the concept and algorithm of the newly developed CGS method are elucidated. Then, 16 practical examples are solved to assess the solving speed of CGS against other iterative methods – GS and modified Gauss–Seidel (MGS, MGS*). The convergence speed of CGS attained 33 times, 3.7 times and 2 times those of GS, MGS and MGS*, respectively.

A dynamic stiffness formulation for the vibration analysis of rotating cross-ply laminated coupled conical–cylindrical–conical shells

In this paper, a dynamic stiffness formulation is reported to analyze the free vibration characteristics of a rotating cross-ply laminated coupled conical–cylindrical–conical shell. In order to ensure the stability of the numerical results, the whole system is firstly divided into substructures, and each substructure is divided into several segments. The theoretical formulation of the rotating shell segment is derived by the first-order shear deformation theory and Hamilton’s principle. The effects of centrifugal force and Coriolis acceleration on the initial hoop tension are considered in the formulation. The dynamic stiffness matrix of the shell segment is derived from the relationship between the state vector and its derivative, where the state vector consists of the resultant force and displacement components of the shell segment. The dynamic stiffness matrix of the whole system is assembled by the displacement coordination relationship. After the number of segments is determined by the convergence analysis, it is compared with the results of the existing literature and finite element software to ensure the accuracy of the dynamic stiffness formulation. Based on this, the effects of rotational speed, geometry and boundary conditions on the structure are studied.

Influence of the Order Exchange of the Node Connection in the Force Analysis of Steel Structures

In the mechanical analysis of steel structures, whether it is static analysis or dynamic analysis, it is necessary to establish the structural stiffness matrix first. In the process of building the structural stiffness matrix, the same element usually has different node code connection orders, and it has never been argued whether the different connection orders of the same element will have an effect on the building of the stiffness matrix. In this study, the influence of the difference in the node connection order on the construction of the element stiffness matrix is studied. First, the structural element stiffness matrix in the global coordinate system is established when the node connection order is different. It is found that the element stiffness matrix in the global coordinate system is indeed inconsistent for the same element with different connection orders. In this study, the elements of the established element stiffness matrix are extracted into the global stiffness matrix of the structural system based on the law of energy conservation; it is found that the global stiffness matrix finally established by using two different connection relationships is the same. The research results of the example show that in the stress analysis of steel structures, selecting different node connection sequences to establish the structural stiffness matrix will obtain the element stiffness matrix under different global coordinate systems. However, through the aggregation process of the global stiffness matrix of the structural system, the global stiffness matrix obtained is consistent, so the different connection sequences of nodes will not affect the stress analysis of steel structures. The example further analyzes the static stress and dynamic responses of the steel structure. The conclusions of this study provide a reliable theoretical basis for the situation that the order of node connections need not be consistent in the finite element modeling of steel structures and are of reference value for the finite element modeling of steel structures.

Analysis of the interior acoustic wave propagation problems using the modified radial point interpolation method (M-RPIM)

Although the radial point interpolation method (RPIM), which is a typical meshless numerical technique, usually behaves much better than the conventional FEM in addressing the numerical dispersion error issue for acoustic computation and more accurate solutions can generally be yielded with the identical node distributions, the related numerical error still can not be completely removed and further improvements are still required. In this work, we proposed a modified RPIM (M-RPIM) to enhance the abilities of the original RPIM in suppressing the numerical dispersion error. In this M-RPIM, a simple and straightforward scheme is employed to ensure that the integrands in performing the numerical integration are continuously differentiable, while in the original RPIM the quadrature cells usually do not align with the shape function supports and then results in the discontinuously differentiable numerical approximation in the quadrature cells, hence considerable numerical integration error can be generated. Since the discontinuously differentiable entities in the system stiffness matrix can be completely avoided in the present M-RPIM, it is found that the numerical dispersion can be markedly suppressed and more accurate numerical solutions can be yielded than the original RPIM in solving acoustic problems. It should be pointed out that the numerical treatments and conclusions in this work are also applicable for most of other meshfree approximations which are similar to the RPIM.

Mooring system stiffness: A six-degree-of-freedom closed-form analytical formulation

A mooring system project is an iterative process, involving design and analysis with numerous parameters. In this context, analytical tools can help to reduce the use of expensive and time consuming full nonlinear numerical simulations based on high-order hierarchical models such as Finite Element models, mainly during early stages of the design. By employing Analytical Mechanics approaches, the present article brings an analytical and closed-mathematical-form formulation to determine the six-degree-of-freedom (DOF) mooring system stiffness matrix at any generic position of the floating body, provided mooring line forces are modeled as conservative ones. In order to demonstrate the use of these tools, the international benchmark of the OC4-DeepCwind platform is considered as a case study. The influence of the vessel mean position and design parameters in the stiffness, natural periods and oscillation modes is studied, revealing interesting coupling effects. Results are benchmarked against the commercial code OrcaFlexTM showing good agreement, observing that the analytical result requires much less computational cost.

Analysis of Sound Propagation Across Acoustic Ducts With Discontinuities Using a High Precision Finite Element Method

A high-precision numerical method is presented to model acoustic wave propagation along a duct with discontinuities. The discontinuous duct can be divided into homogeneous and inhomogeneous substructures. The inhomogeneous substructures are purely discretized by the conventional finite element method, while only cross-sectional areas need to be discretized for homogeneous substructures. The Legendre transformation is then used to transform the semi-discretized problem from the Lagrangian system into the Hamiltonian system. A Riccati equation-based high precision integration method is taken to perform the integral along the longitudinal direction, i.e., the homogeneous direction, to generate stiffness matrices of substructures. The final system stiffness matrix is obtained by assembling stiffness matrices of homogeneous substructures with stiffness matrices of inhomogeneous substructures. Numerical examples are provided to validate the proposed method, and these examples have shown high efficiency and accuracy compared with the conventional finite element method.

Spatial Rigid-Flexible-Liquid Coupling Dynamics of Towed System Analyzed by a Hamiltonian Finite Element Method

An effective Hamiltonian finite element method is presented in this paper to investigate the three-dimensional dynamic responses of a towed cable-payload system with large deformation. The dynamics of a flexible towed system moving in a medium is a classical and complex rigid-flexible-liquid coupling problem. The dynamic governing equation is derived from the Hamiltonian system and built-in canonical form. A Symplectic algorithm is built to analyze the canonical equations numerically. Logarithmic strain is applied to estimate the large deformation effect and the system stiffness matrix will be updated for each calculation time step. A direct integral solution of the medium drag effect is derived in which the traditional coordinate transformation is avoided. A conical pendulum system and a 180° U-turn towed cable system are conducted and the results are compared with those retraced from the existing Hamiltonian method based on small deformation theory and the dynamic software of Livermore software technology corp. (LS-DYNA). Furthermore, a circularly towed system is analyzed and compared with experimental data. The comparisons show that the presented method is more accurate than the existing Hamiltonian method when large deformation occurred in the towed cable due to the application of logarithmic strain. Furthermore, it is more effective than LS-DYNA to treat the rigid-flexible-liquid coupling problems in the costs of CPU time.

Modeling and modal analysis of suspension bridge based on continual formula method

A continnum model is developed for the dynamic analysis of a suspension bridge with inclined main cables and hangers, based on incremental Lagrangian formulation and kinematics theories. The static equilibrium equations are derived from a shape-finding method followed by the equations of motion. Comparison of modal analysis results of the proposed model and the finite element method demonstrates the accuracy of the proposed modelling technique.Modal analysis is further conducted with variations of the following system parameters, i.e. inclination angle of main cable, the ratio of flexural stiffness of deck in the vertical and lateral directions, the ratio of vertical flexural stiffness of deck and tensile stiffness of cable, and torsional moment of inertia of cross-section of deck. Results show that the difference of eigenfrequencies within the eigenfrequency cluster is smaller for a suspension bridge with vertical cables than that with inclined cables. Phenomena of crossover and veering of eigenfrequency loci of the bridge system are observed. The mechanism of such phenomena is further studied with the help of a sectional bridge model. The effect of variation of stiffness matrix of bridge system on the eigenfunction is studied. The condition for crossover and veering of the eigenfrequency loci of this bridge model is identified. It is noted that veering can affect the modal property of the system by coupling two involved modes. This effect should be carefully examined during the design of the suspension bridge.

Multiple bifurcation paths visualized by a computational asymptotic stability theory

Multiple bifurcation (MB) is a compound stability problem of nonlinear structures, in which the singular system stiffness matrix at the stability point coincidentally undergoes two or more zero eigenvalues. The corresponding critical eigenvectors are generally coupled in the actual buckling modes, as frequently observed in symmetric structures.This paper presents a practical diagnosis to visualize all secondary paths branching from the compound stability point when the stiffness matrix is deficient in rank by two. To find post-critical equilibria around an MB point (MBP), asymptotic expansions are assumed to consist of two homogeneous and one particular solution of the singular stiffness equations. Furthermore, hyper-dual numbers are introduced to numerically evaluate the derivatives of the system stiffness with respect to the nodal degrees-of-freedom. The resulting bifurcation equations are a set of three simultaneous cubic polynomial equations with unknown perturbation parameters that can be solved by a popular graphical software. The number and location of existing equilibria above and beneath the MBP at the load level can exactly indicate each type of branching path, such as asymmetric, unstable, or stable symmetric bifurcation paths.Two numerical examples demonstrate that the proposed asymptotic theory can reliably diagnose MB. The first one is the Augusti model, i.e., a simple rigid column supported by elastic springs that shows that the numerical prediction by the proposed method is consistent with Augusti’s analytical results. The second example was computed using the plate and shell finite element (FE) program to verify that the asymptotically expanded and visually solved bifurcation equations work well and can be implemented in existing FE codes for stability analysis, including imperfection-sensitivity and optimization.

Nonlinear analysis of constructions from different materials based on unified plastic constitutive relations

In the paper the plastic constitutive relations based on the hypotheses of the plastic flow theory, unified for different materials, such as soil ground, concrete, metal etc. are proposed. These plastic constitutive relations are based on the hypotheses of the Mises-Schleicher-Botkin theory of failure. The strength characteristics of materials, determined by standard methods, are redefined for octahedral sites, they are invariants for. The problem is solved by the finite element method (FEM). The calculation model MFOE made up of singular finite elements (tetrahedral, triangular plates, rods). This fact increases the accuracy of the description of loading paths in finite elements and ensures a unique correspondence of the constitutive equations to elementary volumes of the construction. The use of singular finite elements is associated with a large amount of access memory for storing the stiffness matrix of the system. So, the range of practically solvable problems limits significantly. To eliminate this contradiction, the Newton’s iteration method – SOR method is developed for equilibrium equations solving in the design of finite elements structures. An iterative algorithm is obtained. It does not require assembly of the system stiffness matrix for its implementation. The volume of operational information is proportional to the number of finite elements in the system. Due to the traditional approach, which requires the assembly of the system stiffness matrix, the amount of operational information is proportional to the square of the degree of kinematic indeterminacy of the system. While using the iterative algorithm, the size of the stiffness matrix and the time of solving are reduced. The results of a nonlinear analysis of soil massifs and concrete structures are represented.

Elastodynamic analysis of multibody systems and parametric mass matrix derivation

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.

Optimization of steel frames with the choice of materials’ grades with restrictions on general and local stability, strength and stiffness

A computational scheme for the optimal design of steel flat frames made of thin-walled rods with closed cross sections has been developed. The structural elements’ total cost is minimized by searching for the materials’ grades and the rods’ cross sections sizes on the variable parameters’ discrete sets. The rods are separately grouped according to the condition of using one steel grade and the criterion for the cross sections’ identity. Active restrictions on the overall structural system’s stability, local stability of the rods’ walls, strength and stiffness are taken into account. The optimization process using a genetic algorithm using a mixed approach to the mutation procedure, a selection option that provides the inoperative design options, and a single-point crossing-over procedure for the simple exclusions from consideration is implemented. The deformable object stress-strain state analysis is carried out on the rod finite element model’s basis. The assessment of ensuring overall stability is carried out by checking the positive definiteness of the tangent stiffness matrix of a finite element system. Local stability using an analytical relationship for rectangular plates, generally subjected to compression-tension and bending in their plane is confirmed. The results of optimization of a three-span frame made of square pipes are presented. Steel grades and profiles were selected for the groups of the structural system’s rods.