System Stiffness Matrix Research Articles

Non-linear modal analysis of structural components subjected to unilateral constraints

In this paper, we present a detailed numerical study of the non-linear dynamics in structural components under unilateral contact constraints. Here, the unilateral term characterises the constitutive law of the restoring force in the constraints as they only sustain elastic reactions in one direction, either compressive or tensile. Thus, the non-differentiability of the contact law at the discontinuity point is the only source of non-linearity. In our approach, the discrete lattice method (DLM) is used to treat the continuous system as a piecewise linear model. Thus, the trajectory of each node in the discrete model would be a sequence of smooth solutions with the switching times between them. The application of the one-step integration scheme allows us to detect the occurrence of contact (i.e. the instants that the lattice nodes cross the discontinuity boundary) and consequently update the active constraints. We also consider embedding the bisection algorithm into the time integration procedure to localise the instants at which the nodes cross the boundary and minimise the accumulative error. Subsequently, the resulting unconditionally stable integration scheme is utilised as the modelling tool in combination with the shooting technique to perform a novel non-smooth modal analysis. In analogy with the smooth non-linear systems, the evolution of non-smooth periodic motions is presented in the frequency-stiffness plots. We apply our method to obtain non-linear normal modes (NNMs) for a number of representative problems, including a bar-obstacle system, a beam-substrate system and a granular chain with tensionless interactions. These numerical examples demonstrate the efficiency of the solution procedure to trace the family of energy-independent non-linear modes across the range of contact stiffnesses. Moreover, the stability analysis of the modes on the plot backbone reveal that they may become unstable due to the interaction with the higher modes or bifurcation of a new NNM with a period equal to the integer multiple of the main mode period. Our results also indicate that the bilinear formula can accurately predict the non-linear frequencies only if the corresponding mode exhibits a smooth character, regardless of the commutativity conditions of the system stiffness matrix. However, it is obvious that the assumption of smooth bilinear behaviour for non-linear modes is not generally valid. This highlights the importance of the present numerical framework for the computation of non-smooth resonance frequencies.

CS-IGA: A new cell-based smoothed isogeometric analysis for 2D computational mechanics problems

In this work a new cell based strain smoothing formulation for isogeometric finite element analysis is proposed. The new CS-IGA method is formulated by incorporating cell-wise strain smoothing operation into IGA method. The compatible strain fields are smoothed based on smoothing domains associated with entities of isogeometric elements and the smoothed Galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix. Unlike IGA, this new approach avoids gradient computation and the assembling of stiffness matrices. A transformation method is employed to accurately solve the problem and enforce properly the essential boundary conditions. Several numerical examples are investigated to verify the validity and accuracy of the proposed method. It has been found that numerical results obtained from CS-IGA can achieve very satisfactory accuracy and improved convergence rate.

Nonlinear vibration of marine riser with large displacement

This paper investigates the nonlinear free vibration of a marine riser with large displacement. The nonlinear equation of motion is derived using a variational approach based on Hamilton’s principle. The strain energy functional is composed of the strain energies due to axial deformation and bending. The strain energy associated with the rotational restraint at the bottom end of riser is also considered. The kinetic energy is derived by considering the riser and transporting fluid motions. The finite element method is applied to develop the nonlinear stiffness matrix, mass matrix, and gyroscopic matrix of the riser system. Consequently, the nonlinear natural frequencies and their corresponding mode shapes are determined by solving the eigenvalue problem incorporating the direct iteration technique. Various numerical examples are investigated to evaluate the linear and nonlinear dynamic characteristics of the riser. The parametric investigations of the horizontal top end offset, top tension, and rotational spring constant at bottom support are provided to show the modal transition phenomenon of the riser system. The nonlinear natural frequencies and their corresponding mode shapes are also presented to identify the softening and hardening behaviors of the riser.

Inelastic parametric analysis of frequencies and seismic responses of soil-multistorey bidirectional eccentric structure interaction system

This paper presents a parametric study on the lateral-torsional coupling behaviours of soil-multistorey bidirectional eccentric structure interaction system from elastic stage to nonlinear inelastic stage. A simplified bidirectional eccentric multistorey dual lateral load-resisting model for the superstructure is proposed to conduct this parametric study. Incorporating the inter-storey restoring force model of the superstructure, the dynamic motion equations and the dynamic stiffness matrix of the soil-multistorey bidirectional eccentric structure interaction system in different loading stages are derived. Based on these, parametric analyses of frequencies and seismic responses for the soil-three storey structure interaction systems from elastic stage to nonlinear inelastic stage are evaluated. Effects of the uncoupled torsion to lateral frequency ratios (Ω) of the corresponding fixed base system, the stiffness eccentricities and different soil types on the frequencies and seismic responses are investigated. Results show that the soil can significantly reduce the lateral-torsional coupling effect and the structural seismic response of the soil-bidirectional eccentric structure interaction system; a smaller Ω value and larger values of the eccentricities lead to a larger reduction in the lateral-torsional coupling effect of the first order mode in nonlinear inelastic stage; the effects of different foundation conditions on the structural seismic responses are weakened by the development of structural inelastic; after considering the soil-structure interaction, the influences of Ω and eccentricities on the structural seismic responses are changed. A softer soil-structure interaction system has a more weaken lateral-torsional coupling effect, and its vibration modes in nonlinear inelastic stage are changed compared with the system situated on other soils.

Feasibility investigation of large-scale model suspended by cable-driven parallel robot in hypersonic wind tunnel test

Cable-driven parallel robot is a special kind of robot, which is actuated by cables. It is already applied in the low speed wind tunnel to get aerodynamic measurement of aircraft model, and the aircraft pose could be adjusted by changing the cable length. Whether it can be used in hypersonic wind tunnel still needs further discussion. This paper presents the dynamics and aerodynamics analysis of a large-scale model supported by 6-DOF cable-driven parallel robot to investigate the feasibility of this special kind of suspension system in hypersonic wind tunnel. The description of this setup with a X-51A-like model is given, and then based on the system dynamic equations, aerodynamic force and stiffness matrix are derived. In the simulation, properties of dynamics and aerodynamics are mainly concerned. A typical shock tunnel with flow duration of about 100 milliseconds is taken as an example, and results show that the system is stable enough to meet the fundamental static wind tunnel test. From the cable tension variation under impact load and the sensitivity analysis, it is likely accessible to derive the aerodynamic forces. Compared with the sting suspension method, cable-driven parallel robot has the priority of higher inherent frequency and more flexible degrees. The interference to the flow field induced by cables is also preliminarily proved to be small by the CFD simulation, which can be acceptable and corrected. Researches conducted show the feasibility of cable-driven parallel robot’s application in hypersonic wind tunnel.

Three-dimensional thermo-hydro-mechanical responses of stratified saturated porothermoelastic material

An exact solution for three-dimensional thermo-hydro-mechanical responses of stratified saturated porothermoelastic material containing a heat source is derived in this paper. Starting with the governing equations of porothermoelastic materials in the Cartesian coordinates, analytical layer-elements of a finite layer and a half-space are obtained in the transformed domain with the aid of Laplace–Fourier transform. The global stiffness matrix of a multi-layered system is assembled and solved in the transformed domain by considering the continuity conditions at adjacent layer interfaces and the boundary conditions. By introducing the inversion of Laplace–Fourier transform, the actual solution in the physical domain is obtained. Finally, a numerical example is carried out to confirm the efficiency and accuracy of the present method, and then a series of numerical examples are performed to study the effects of heat source's half-life and shape, thermo-permeability ratio and stratified material properties on the responses of saturated porothermoelastic material.

Edge-Based Smoothed Three-Node Mindlin Plate Element

AbstractThe edge-based smoothed finite element method (ES-FEM) was proposed recently to improve the accuracy of the standard finite element method for solid mechanics. In the present paper, the ES-FEM is incorporated with the three-node Mindlin plate element (MIN3) to give a novel edge-based smoothed MIN3 (ES-MIN3) for plate analysis. The system stiffness matrix is computed by employing the edge-based strain smoothing technique over the edge-based smoothing domain. For the purpose of avoiding the transverse shear-locking, the MIN3 element is performed to calculate the strains in each element. From a series of selected numerical examples, it is found that the present ES-MIN3 possesses highly accurate solutions, and can be competitive with many existing plate elements.

A variational formulation for vibro-acoustic analysis of a panel backed by an irregularly-bounded cavity

A weak form variational based method is developed to study the vibro-acoustic responses of coupled structural-acoustic system consisting of an irregular acoustic cavity with general wall impedance and a flexible panel subjected to arbitrary edge-supporting conditions. The structural and acoustical models of the coupled system are formulated on the basis of a modified variational method combined with multi-segment partitioning strategy. Meanwhile, the continuity constraints on the sub-segment interfaces are further incorporated into the system stiffness matrix by means of least-squares weighted residual method. Orthogonal polynomials, such as Chebyshev polynomials of the first kind, are employed as the wholly admissible unknown displacement and sound pressure field variables functions for separate components without meshing, and hence mapping the irregular physical domain into a square spectral domain is necessary. The effects of weighted parameter together with the number of truncated polynomial terms and divided partitions on the accuracy of present theoretical solutions are investigated. It is observed that applying this methodology, accurate and efficient predictions can be obtained for various types of coupled panel-cavity problems; and in weak or strong coupling cases for a panel surrounded by a light or heavy fluid, the inherent principle of velocity continuity on the panel-cavity contacting interface can all be handled satisfactorily. Key parametric studies concerning the influences of the geometrical properties as well as impedance boundary are performed. Finally, by performing the vibro-acoustic analyses of 3D car-like coupled miniature, we demonstrate that the present method seems to be an excellent way to obtain accurate mid-frequency solution with an acceptable CPU time.

Structural Damage Detection by Using Single Natural Frequency and the Corresponding Mode Shape

Damage can be identified using generalized flexibility matrix based methods, by using the first natural frequency and the corresponding mode shape. However, the first mode is not always appropriate to be used in damage detection. The contact interface of rod-fastened-rotor may be partially separated under bending moment which decreases the flexural stiffness of the rotor. The bending moment on the interface varies as rotating speed changes, so that the first- and second-modal parameters obtained are corresponding to different damage scenarios. In this paper, a structural damage detection method requiring single nonfirst mode is proposed. Firstly, the system is updated via restricting the first few mode shapes. The mass matrix, stiffness matrix, and modal parameters of the updated system are derived. Then, the generalized flexibility matrix of the updated system is obtained, and its changes and sensitivity to damage are derived. The changes and sensitivity are used to calculate the location and severity of damage. Finally, this method is tested through numerical means on a cantilever beam and a rod-fastened-rotor with different damage scenarios when only the second mode is available. The results indicate that the proposed method can effectively identify single, double, and multiple damage using single nonfirst mode.

A stable node-based smoothed finite element method for acoustic problems

It is well-known that the classical “overly-soft” node-based smoothed finite element method (NS-FEM) fails to provide reliable results to the Helmholtz equation due to the “temporal instability”. To cure the fatal drawback of NS-FEM and reduce the dispersion error in computational acoustics, this paper proposed a stable node-based smoothed finite element method (SNS-FEM) for analyzing acoustic problems using linear triangular (for 2D space) and tetrahedral (for 3D space) elements that can be generated automatically for any complicated configurations. In the present formulation, the system stiffness matrix is computed using the smoothed acoustic pressure gradients together with the gradient variance items over the smoothing domains associated with nodes of element mesh. It turns out the addition of stabilization term makes the SNS-FEM possess an ideal stiffness, thus successfully cures the temporal instability and significantly reduces the dispersion error in acoustic problems. Numerical examples, including both benchmark cases and practical engineering problems, demonstrate that the SNS-FEM possesses the following important properties: (1) temporal stability; (2) super accuracy and super convergence; (3) higher computational efficiency; (4) insensitive to mesh distortion.

Helicopter Blade Stability Analysis Using Aeroelastic Frequency Response Functions

In the present paper, the aeroelastic stability of helicopter rotor blade is determined using Aeroelastic Frequency Response Function. The conventional methods of aeroelastic stability usually use an iterative procedure while the present method does not require such approach. Aeroelastic Frequency Response Functions are obtained by inverting dynamic stiffness matrix of the aeroelastic system. System response could be obtained through exciting each degree of freedom. The resulting response was then plotted and the behavior of this function was investigated to find out the stability criteria and system natural frequencies. The results of this method are compared with stability boundaries obtained from the conventional p-k method and it can be inferred that, compared to other methods, the present algorithm is of less numerical cost.

Decomposed Interval Matrix Perturbation Finite Element Method for the Response Analysis of the Acoustic Field with Uncertain Parameters

To further improve the computational accuracy and efficiency of the modified interval perturbation finite element method(MIPFEM),a decomposed interval matrix perturbation finite element method(DIMPFEM) is proposed. In the proposed method,the dynamic stiffness matrix of an acoustic system is decomposed into the sum of several sub-matrices whose perturbation matrix can be expressed as the products of perturbation factors and determine matrices,thus the errors arising from the first-order Taylor expansion can be avoided. To achieve a higher computational efficiency,the inverse perturbation matrix,approximated by the modified Neumann series expansion is calculated by the epsilon-algorithm. Numerical examples on a 2D acoustic tube and a 2D acoustic cavity of a multi-purpose vehicle(MPV) with interval parameters verify that the computational accuracy and efficiency of DIMPFEM are higher than those of MIPFEM.

A systematic approach for precision electrostatic mode tuning of a MEMS gyroscope

In this paper a systematic approach to precision electrostatic frequency tuning of the operational modes of a MEMS ring vibratory gyroscope is presented. In both rate and rate integrating gyroscopes the frequency split between the two modes of vibration which detect the Coriolis acceleration is one of the principal factors in determining the sensitivity and noise floor of the sensor. In high precision applications in the defence/aerospace sector a frequency split of the order of 10 mHz or less is highly desirable. In the ground-breaking Hemispherical Resonator Gyroscope (HRG) electrostatic tuning has been employed as a tuning mechanism. However, a description of the procedure is not available in the literature. The tuning scheme described here involves assessing mode mistuning by the ratio of the in-phase and quadrature components of the response to an external force that has similar properties to the gyroscopic force resulting from Coriolis action, and choosing the tuning voltages so that independent modification of the elements of the system stiffness matrix can be achieved. Experiments on a commercially available gyroscope with a natural frequency of 14 kHz show that the frequency split can be reduced from 1.5 Hz to 6 mHz. This represents a frequency precision of better than 1 part in a million.

Design and Application of Quadrature Compensation Patterns in Bulk Silicon Micro-Gyroscopes

This paper focuses on the detailed design issues of a peculiar quadrature reduction method named system stiffness matrix diagonalization, whose key technology is the design and application of quadrature compensation patterns. For bulk silicon micro-gyroscopes, a complete design and application case was presented. The compensation principle was described first. In the mechanical design, four types of basic structure units were presented to obtain the basic compensation function. A novel layout design was proposed to eliminate the additional disturbing static forces and torques. Parameter optimization was carried out to maximize the available compensation capability in a limited layout area. Two types of voltage loading methods were presented. Their influences on the sense mode dynamics were analyzed. The proposed design was applied on a dual-mass silicon micro-gyroscope developed in our laboratory. The theoretical compensation capability of a quadrature equivalent angular rate no more than 412 °/s was designed. In experiments, an actual quadrature equivalent angular rate of 357 °/s was compensated successfully. The actual compensation voltages were a little larger than the theoretical ones. The correctness of the design and the theoretical analyses was verified. They can be commonly used in planar linear vibratory silicon micro-gyroscopes for quadrature compensation purpose.

Coupling Analysis of Liquid Sloshing and Structural Vibration Using General Software

In this paper, a convenient modal analysis method for the linear coupled vibration of a container that is partially filled with a fluid is introduced. This problem is important for various reasons, such as stability analysis. The fluid-structure interactions in an elastic tank with an incompressible liquid are assumed to produce small vibrations. Reduced symmetric finite element equations of the system are acquired according to the component mode synthesis method. Considering that the liquid satisfies the same governing equation as steady heat conduction, general programs can be used to calculate the mass matrix and stiffness matrix of the coupled system. Then, modal analysis of the liquid container using general software, e.g., MSC Nastran, that ensures accuracy and stableness in the process, is applied to demonstrate that this method can determine the modal frequency in a fluid-structure coupled system.

A cell-based smoothed three-node Mindlin plate element (CS-FEM-MIN3) based on the C0-type higher-order shear deformation for geometrically nonlinear analysis of laminated composite plates

A cell-based smoothed three-node Mindlin plate element (CS-FEM-MIN3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and free vibration analyses of Mindlin plates. In this paper, the CS-FEM-MIN3 is extended to geometrically nonlinear analysis of laminated composite plates. The nonlinear formulation is based on the C0-type high-order shear deformation plate theory (C0-HSDT) and the von Kármán strains, which deal with small strains and moderate rotations. In the process of formulating the system stiffness matrix of the CS-FEM-MIN3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the MIN3 is used to compute the strains. Then the strain smoothing technique on whole the triangular element is used to smooth the strains on these three sub-triangles. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of available other numerical results.

Dynamic response of pavement under FWD using spectral element method

This paper presents the development of Spectral Element Method (SEM) for simulating pavement response under FWD pulse loads. Based on Fourier-Bessel function and Fast Fourier Transform, the algorithms and procedures of SEM are introduced, including the formation of layered stiffness matrix and system stiffness matrix with different bond conditions between two interfaces. Deflections calculated from the SEM under different pulse durations are compared and verified with static deflections calculated from BISAR3 and measured dynamic deflections separately. Research results indicate that pavement deflections are sensitive to load pulse duration. The SEM can also be used to solve viscoelastic problems by choosing proper rheological models.

Immersed Finite Element Method for Interface Problems with Algebraic Multigrid Solver

AbstractThis article is to discuss the bilinear and linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. For the numerical methods based on finite difference formulation and a structured mesh independent of the interface, the stiffness matrix of the linear system is usually not symmetric positive-definite, which demands extra efforts to design efficient multigrid methods. On the other hand, the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite. Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems. The numerical examples demonstrate the features of the proposed algorithms, including the optimal convergence in both L2and semi-H1norms of the IFE-AMG solutions, the high efficiency with proper choice of the components and parameters of AMG, the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems, and the relationship between the cost and the moving interface location.

Nonlinear structural modification and nonlinear coupling

Structural modification methods were proved to be very useful for large structures, especially when modification is local. Although there may be inherent nonlinearities in a structural system in various forms such as clearances, friction and cubic stiffness, most of the structural modification methods are for linear systems. The method proposed in this work is a structural modification/coupling method developed previously, and extended to systems with nonlinear modification and coupling. The method is most useful for large linear structures with nonlinear local modification or coupling. It is based on expressing nonlinear internal force vector in a nonlinear system as a response level dependent “equivalent stiffness matrix” (the so-called “nonlinearity matrix”) multiplied by the displacement vector, through quasilinearizing the nonlinearities using describing functions. Once nonlinear internal force vector is expressed as a matrix multiplication form then several structural modification and/or coupling methods can easily be used for nonlinear systems, provided that an iterative solution procedure is employed and convergence is obtained. In the proposed approach the nonlinear FRFs of a modified/coupled system are calculated from those of the original system and dynamic stiffness matrix of the nonlinear modifying system. Formulations and sample applications of the proposed approach for each of the following cases are given, nonlinear modification of a linear system with and without adding new degrees of freedom, and elastic coupling of a nonlinear subsystem to a main linear system with linear or nonlinear elements. Case studies are given for the verification of the method and then a real life application of the method is presented.

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise theory based on the C0-HSDT for analyses of composite plates

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) based on the first-order shear deformation theory (FSDT) using triangular element was recently proposed for static and dynamics analyses of Mindlin plates. In this paper, the CS-FEM-DSG3 is extended and incorporated with a layerwise theory for static and free vibration analyses of composite and sandwich plates. In the layerwise theory, the behavior of each layer follows the C0-type higher-order shear deformation theory (C0-HSDT) and the condition of displacement continuity is imposed at the interfaces of layers. The shear correction factor is hence no more necessary and the accuracy of transverse shear stresses is improved significantly. In the process of formulating the system stiffness matrix of the CS-FEM-DSG3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the stabilized discrete shear gap method (DSG3) is used to compute the strains. Then the strain smoothing technique on whole the triangular element is used to smooth the strains on these three sub-triangles. Some numerical studies have been conducted to demonstrate the efficient performance of the proposed formulation.