Global Dynamic Stiffness Matrix Research Articles

Exact closed-form dynamic stiffness matrix of damaged frames comprising Timoshenko–Ehrenfest beams

The present work proposes the extension of an exact distributional model for cracked beams to include the local influence on axial deformation on shear deformable beams. Axial, shear and flexural concentrated flexibilities are taken into account to model the damaged sections. Shear deformability is accounted for by adopting the Timoshenko–Ehrenfest model and exact closed-form solution of the mode shapes for multiple cracked beams are derived. On this ground, the final achievement of the work consists in the original formulation of the Dynamic Stiffness Matrix (DSM) with multiple cracked sections in exact closed form. The closed-form expression of the DSM allows the assemblage of the global DSM of a structure to conduct the free vibration analysis of damaged framed structures as well as for the study of frames with semi-rigid joints. This approach has the advantage to avoid any numerical procedure for the construction of the DSM as well as the introduction of further degrees of freedom (dofs) in correspondence of the damaged sections, keeping the size of the problem the same as in the intact configuration. Natural frequencies of the frames are determined by means of the Wittrick–Williams algorithm (W–W), then the mode shapes and modal forces are subsequently obtained in closed form. The proposed approach is validated with results available in literature for two portal frames characterized by the presence of cracks and semi-rigid joints, respectively. Finally, the potentiality of the approach, also in view of practical applications, is demonstrated analysing a multi-storey frame with semi-rigid connections.

Distributed Transfer Function Formulations for Vibration Analysis of Cracked Beam and Frame Structures

In this paper, based on distributed transfer function method (DTFM), the closed-form analytical solutions for vibrations of Euler–Bernoulli beam and frame structures with arbitrary number of cracks are studied. First, generalized DTFM is employed to characterize the dynamical model for a single cracked beam and its analytical solutions for eigenvalue problem and frequency response are obtained. Then, a new DTFM cracked element that encapsulates one crack of arbitrary location inside the beam is proposed. Using the DTFM cracked element and global dynamic stiffness matrix assembly technique, damaged frame structures of arbitrary form can be modeled for vibration analysis. Previous analytical methods only addressed low-frequency vibration of simple cracked beam structures, the proposed method can yield analytical solutions in the medium- and high-frequency regions, which is critical for the small crack detection in complex frames. Lastly, three numerical examples are given to illustrate the correctness and effectiveness of the DTFM in analyzing natural frequencies, modal shapes and frequency responses for cracked structures. By comparing with the Finite Element Method (FEM) and benchmarks from literatures, we proved that the DTFM has better performances in terms of accuracy and efficiency.

Vibration response of FG-CNTRC plate resting on non-homogeneous elastic foundation via a closed-form dynamic stiffness formulation

In this work, a closed-form dynamic stiffness matrix has been formulated to study the influence of various configurations of non-homogeneous elastic foundation on the free vibration characteristics of carbon nanotube-reinforced functionally graded (FG-CNTRC) plate. The plate kinematic variables are defined based on the first order shear deformation theory. Through the application of Hamilton’s principle, frequency dependent dynamic stiffness (DS) matrix is formulated for a Levy type plate element supported on the Winkler-Pasternak foundation. Appropriate assembly of these DS matrices leads to the formation of global dynamic stiffness matrix, for the non-homogeneously supported plate, which is finally solved using the well-established Wittrick–Williams algorithm to compute the mode shapes and natural frequencies to a desired accuracy. Thus, the present work exploits the applicability of the dynamic stiffness method (DSM) for the free vibration study of the FG-CNTRC plate supported by partial and non-homogeneous elastic foundations. A comparative study is also carried out to ascertain the exactness and the computational efficiency of the dynamic stiffness method. Results from the parametric studies show that the non-homogeneous supports greatly influence the frequency as well as the modeshape of the plate. It is observed that the Pasternak parameter has a profound effect on the response of the plates as compared to that of the Winkler parameter. Effect of different CNT distributions on the frequency behavior are also discussed in detail. It can be emphasized that the new set of DSM computed results reported in the paper for diverse plate configurations may serve as reference results for the future research studies.

Wave propagation in periodic nano structures through second strain gradient elasticity

The study of wave propagation in the periodic nano-waveguides is of importance to advance the development of energy and wave controlling systems. In this paper, firstly, the element discretization of nano structures is introduced based on the second strain gradient theory. The displacement field is determined through the utilization of six quintic Hermite polynomial interpolating functions. By applying Hamilton’s principle, the weak form, which incorporates the element matrices, is determined and the global dynamic stiffness matrix of a unit cell is assembled. Then, the wave finite element method, based on the inverse form, direct form and contour integral solution, is used to analyze the two-dimensional free wave propagation properties of complex nano structures by solving the eigenvalue problems. The interpretation of the frequency spectrum is confined to the irreducible first Brillouin zone, encompassing both the dispersion relation and band structure. Furthermore, the slowness surfaces and energy flow vector fields are discussed via the second strain gradient theory. The novel work, which utilizes the second strain gradient theory equipped with the wave finite element method to investigate two-dimensional complex nano structures, has revealed significant potential in elucidating the two-dimensional wave propagation characteristics of the complex nano-waveguides.

An efficient three-dimensional dynamic stiffness-based model for predicting subway train-induced building vibrations

The prediction of train-induced vibration in buildings constructed above the subway is a challenging problem due to its sophisticated system and time-consuming calculation. This paper proposes an efficient three-dimensional (3D) dynamic stiffness-based model (DSM) to predict subway train-induced building vibrations. The DSM has the capability to analytically explore the axial, bending, and torsional vibrations of columns and beams, as well as the flexural and in-plane vibrations of floors and walls induced by the moving train, which few studies have focused on before, and has high computational efficiency compared to the traditional finite element method (FEM). In the DSM, the dynamic stiffness matrices of floors, walls, columns and beams are formulated by the stiffness theory firstly. Then, the global dynamic stiffness matrix of the building is determined following an assembly procedure and the vibration responses of the building can be calculated by the dynamic stiffness equation in the frequency domain. Further, the calculation accuracy and efficiency of the DSM are validated by comparison with a 3D finite element model. Finally, the DSM is used to predict the train-induced vibrations of a building constructed at a subway station based on a subway train-slab track coupled dynamics model. Results show that the building vibrations induced by the moving train are significant in both vertical and horizontal directions and the steel-spring floating-slab track can effectively suppress the train-induced building vibrations.

Dynamic response of a multi-scale layered saturated porous half-space due to seismic dislocation source by using a revised dynamic stiffness matrix method

The study investigating the dynamic response of seismic dislocation source in a layered saturated porous half-space, particularly the multi-scale characteristic of half-space, from crustal to geotechnical scale, is still lacking in the literature. particularly in will lead to the computation bottlenecks for multi-scale seismic wave propagation. Based on Biot's theory of wave propagation in a fluid-saturated porous solid, a dynamic stiffness matrix method is proposed to investigate the dynamic response of a multi-scale layered saturated porous half-space due to the seismic dislocation source. Assembling each layer stiffness matrix in the frequency-wavenumber domain, the global dynamic stiffness matrix is constructed to avoid errors accumulative compared to the reflection and transmission matrix method. The divergent index terms of the stiffness matrix with respect to thickness and wavenumber are extracted to revise the dynamic stiffness matrix adapted for multi-scale half-space. The seismic dislocation source action is equal to the external forces imposed on the total saturated porous half-space, which can obtain the dynamic response by using revised dynamic stiffness matrix method. The accuracy of the proposed method is verified via the comparison between the calculated results and of three published literatures. The multi-scale model with realistic near-surface fine structure is selected to investigate the effects of superficial fine structures on the dynamic responses. The numerical simulations results show that the presence of low-velocity saturated porous layers has a prominent impact on ground motion. The dynamic response seems to be sensitive to the porosity conditions, especially for high frequency, which cause the difference in variation of peak displacement value.

Frequency domain-based analytical framework for seismic performance of viscously damped outrigger systems based on continuous Timoshenko beam theory

This paper proposes a frequency domain-based analytical framework for seismic performance of viscously damped outrigger systems. Based on a core–outrigger–damper–column simplified model, the global dynamic stiffness matrix is assembled from the core modeled as a Timoshenko beam, damped outriggers as complex rotational stiffness comprising outriggers, dampers, and perimeter columns, and inherent damping using Leung’s theory and modal damping construction. A general numerical method combining Wittrick-Williams algorithm and Newtonian iteration is developed to study the dynamic characteristic of such systems with multiple damped outriggers. The fast Fourier transformation (FFT) and the inverse Fourier transformation (IFFT) are then integrated with the principle of potential energy to obtain the equivalent nodal force and thus the time history response by transformations between time and frequency domains. Finally, the stochastic analysis is conducted via the transfer function resulting from the global stiffness matrix with the stochastic seismic excitation following Kanai-Tajimi spectrum. The proposed approach is verified by comparison with the finite element method through a case study of a tall building implemented with viscously damped outriggers. This study shows that the proposed analytical framework could serve as a powerful tool for evaluating the performance of viscously damped outrigger systems.

Multi-mode propagation and diffusion analysis using the three-dimensional second strain gradient elasticity

The multi-mode propagation and diffusion properties are crucial informations when studying complex waveguides. In this paper, firstly, the three-dimensional modeling of micro-sized structures is introduced by using the second strain gradient theory. The constitutive relation is deduced while the six quintic Hermite polynomial shape functions are employed for the displacement field. The weak formulations including element stiffness and mass matrices and the force vector are calculated through the Hamilton’s principle and the global dynamic stiffness matrix of a unit cell is assembled. Then, free wave propagation characteristics are analyzed by solving eigenvalue problems within the direct wave finite element method framework. The dispersion relations of positive going waves considering the size effects are illustrated. Furthermore, the effects of higher order parameters on the dispersion curves are discussed and the forced responses with two boundary conditions are expounded. Eventually, the wave diffusion including reflection and transmission coefficients are illustrated through simple and complex coupling conditions, respectively. The dynamic analysis of coupled waveguides through the wave finite element method equipped with the second strain gradient is a novel work. The results show that the proposed approach is of significant potential for investigating the wave propagation and diffusion characteristics of micro-sized structures.

Vibration Characteristic Analysis of Tunnel Structure Buried in Elastic Foundation Based on Dynamic Stiffness Matrix

In this study, the free vibration characteristics of a non-circular tunnel buried in a Winkler foundation are first investigated based on a dynamic stiffness approach and the Wittrick–Williams algorithm. The non-circular tunnel was modeled as a shell composed of multipartial circular cylinders based on the first shear deformation Flügge shell theory, by which the effects of shear deformation and moment of inertia were considered. The transfer function of the state variable was derived, and the elementary dynamic stiffness matrices were formulated based on this transfer function. Similar to the finite element method, the global dynamic stiffness matrix was established by assembling each elementary dynamic stiffness matrix. By solving the global dynamic stiffness matrix using the Wittrick–Williams algorithm, the natural frequencies and the corresponding mode shapes were determined. Subsequently, the calculated results were compared with literature and finite element analysis results, thus validating the accuracy and reliability of the proposed method. A parametric analysis was conducted to investigate the effects of the cross-section, stiffness coefficient, thickness, and length of the tunnel composed of multipartial circular cylinders. It was concluded that among the four types of cross-sections, the free vibration characteristics of the shells with close isoperimetric radii are similar. The natural frequencies of the shells buried in a moderately stiff Winkler foundation increase rapidly with its stiffness coefficient, whereas those of the shells buried in softer and stiffer foundations remain practically constant. The increase in shell thickness progressively increases the natural frequencies. In contrast, the natural frequencies decrease sharply with the increase in shell length; they first decrease to the lowest limits and subsequently remain unaltered. Thus, the free vibration characteristics of an actual tunnel structure resting on a Winkler foundation can be obtained by conducting the free vibration analysis on a sufficiently long shell segment, which is of major significance in the dynamic design of tunnel structures.

Dynamic Stiffness Matrix Approach to Free Vibration Analysis of Functionally Graded Rotor Bearing System Subjected to Thermal Gradients.

The dynamic stiffness matrix (DSM) method, an analytical method that provides exact solutions, has been used for the first time for the free vibration analysis of a functionally graded (FG) rotor bearing system subjected to temperature gradients and to investigate its application to FG rotors. The material gradation occurs based on the power law between the inner metal core and the outer ceramic rich layer of the FG rotor. The temperature gradation follows the Fourier law of heat conduction which leads to non-linear temperature distribution (NLTD) in the radial direction of the FG rotor. The development of the DSM formulations for Timoshenko FG rotor elements using the governing equations derived from translational and rotational equilibrium conditions is the novelty of the present work. The DSM of the FG rotor elements, rigid disk and linear isotropic bearings are assembled to obtain the global dynamic stiffness matrix of the FG rotor bearing system. The natural whirl frequencies are computed from the global DSM using the Wittrick–William algorithm as a root searching technique. The natural and whirl frequencies are validated with the results available in the literature and the exactness of the DSM method has been exemplified.

The dynamic stiffness matrix method for seismograms synthesis for layered transversely isotropic half-space

Natural rocks and soils show significant anisotropic behavior owing to sedimentation and weathering, particularly apparent in the crust. However, seismograms synthesis for the anisotropic medium is scarce, especially bottlenecks for large-scale and broadband seismic wave propagation. In this paper, seismogram synthesis for a multi-layered transversely isotropic half-space due to seismic point sources is addressed by using a dynamic stiffness matrix method. The critical step of the proposed method lies in the construction of the global dynamic stiffness matrix in frequency-wavenumber domain, which avoids the accumulation of errors that appeared in the reflection and transmission matrix method. Exponential divergent terms related to wavenumber and layer thickness in dynamic stiffness matrix are extracted for fast simulating large-scale and high-frequency wavefield. The seismic point sources are equated to the external loads imposed at the introduced auxiliary surfaces, through which the dynamic stiffness matrix can be utilized to obtain the solutions. The accuracy is verified by comparing the proposed method results with three examples in the publications, involving different source types and depth scales. Two numerical examples regarding Thomsen parameters are given to illustrate the mechanism of wave propagation and ground motion in an anisotropic media. Numerical results show that the anisotropic characteristics affect the wave propagation time, waveform and wave-front energy distribution, further leading to shear wave splitting phenomena, which remarkably differ from the isotropic case. A case study using a simplified three-layered crustal structure model in Tibet elucidates that peak ground displacement and its distribution difference within the epicenter of 10 km by comparing with the equivalent isotropic model without considering anisotropic properties. The results show a 10.91% deviation in the location of the peaks, with amplitude differences of up to 120% due to a 45° dip-slip source at different frequencies (from 1 Hz to 15 Hz).

Dynamic stiffness formulation and vibration analysis of coupled conical-ribbed cylindrical-conical shell structure with general boundary condition

In this paper, the dynamic stiffness method (DSM) is presented to investigate the vibration characterizes of coupled conical-ribbed cylindrical-conical shell structure with general boundary condition. Rib stiffeners are evenly distributed in the inner side of the cylindrical shell. Firstly, the coupled structure is divided into several components (i.e., conical and cylindrical shells, annular plate) according to the geometric properties and coupling boundary conditions. The displacement fields of the components are derived from the governing equations exactly. Then, the dynamic stiffness matrix is formulated based on the relationship between the displacements and forces on the boundaries of the components. The global dynamic stiffness matrix of the coupled structure is assembled in a similar way as used in the finite element method (FEM). In addition, the experimental model of conical-ribbed cylindrical-conical shell structure is built and vibration test is conducted. The accuracy and stability of the present method are validated by comparing the present results with the results obtained by FEM and experimental test. Several numerical examples are conducted to understand the effects of geometric parameters on the dynamic behaviors of the coupled structure, which can offer positive guides for acoustic design of the combined shell structure.

On the dynamics of nano-frames

Size-dependent dynamic responses of small-size frames are modelled by stress-driven nonlocal elasticity and assessed by a consistent finite-element methodology. Starting from uncoupled axial and bending differential equations, the exact dynamic stiffness matrix of a two-node stress-driven nonlocal beam element is evaluated in a closed form. The relevant global dynamic stiffness matrix of an arbitrarily-shaped small-size frame, where every member is made of a single element, is built by a standard finite-element assembly procedure. The Wittrick–Williams algorithm is applied to calculate natural frequencies and modes. The developed methodology, exploiting the one conceived for straight beams in [International Journal of Engineering Science 115, 14–27 (2017)], is suitable for investigating free vibrations of small-size systems of current applicative interest in Nano-Engineering, such as carbon nanotube networks and polymer-metal micro-trusses.

The application of the Wittrick-Williams algorithm for free vibration analysis of cracked skeletal structures

In this study, the dynamic stiffness method is applied to a cracked beam that incorporates Rayleigh bar theory and Timoshenko beam theory, and the Wittrick-Williams algorithm is used to pinpoint natural frequencies. The influence of a crack on the two components of the Wittrick-Williams algorithm, namely, the number of negative leading diagonal terms in the upper triangular form of the global dynamic stiffness matrix at a trial frequency, and the sum of frequencies of each clamped-clamped element of the structure exceeded by the trial frequency, is explored. For the first component, the leading diagonal terms relevant to the additional degrees of freedom introduced by the crack should be considered. While for the second component, it is shown that a crack element itself does not contribute to the second component. The influence of a crack element on the second component only lies in that the crack alters the discretization thus more beam elements are involved. The same philosophy applies to the introduction of an affiliated mass or spring-mass system. To verify this statement, the free vibration of a cracked single-storey frame carrying a roving mass with rotary inertia is studied. The frequency results obtained using the dynamic stiffness method incorporating the Wittrick-Williams algorithm are compared with those exported from a finite element model in ANSYS. Some discussions about the results are made. The effects of roving translational inertia and roving rotary inertia are studied.

Vibration analysis of combined conical-cylindrical shells based on the dynamic stiffness method

An analytic dynamic stiffness method (DSM) is extended to analyze the free and forced vibration of combined conical-cylindrical shells with general boundary conditions in this paper. Flügge shell theory is utilized to formulate the motion equations of each shell component. Based on the exact general solutions of the motion equations, the dynamic stiffness matrices for each shell segment are established separately and the global dynamic stiffness matrix for the whole structure is established by assembling the dynamic stiffness matrices for each shell segment similarly as finite element method (FEM) did. Natural frequencies and forced responses of the combined shells are obtained based on the global dynamic stiffness matrix. Through comparing vibration results of DSM with ones from open literature and FEM, rapid convergence, good accuracy, and high efficiency of the DSM are demonstrated. In the numerical examples, the influences of boundary conditions, axial and circumferential mode numbers, and semi-vertex angles on the free vibration are studied. The effects of direction and location of external force and structural damping on the forced vibration are also discussed.

Exact free vibration analysis for membrane assemblies with general classical boundary conditions

This paper proposes exact dynamic stiffness formulations for membranes and their assemblies under any arbitrary classical boundary conditions. First, by taking exact solutions in one direction satisfying all possible opposite edge supports, we can derive exact general solutions of the Helmholtz equation for membrane vibration. Then, generic force and displacement boundary conditions in the other direction are expressed in terms of the general solutions. Finally, the dynamic stiffness matrices of rectangular membrane elements are formulated, which can be assembled directly and allows applications of arbitrary boundary conditions. As an accurate and efficient modal solution technique, the Wittrick-Williams (WW) algorithm is applied onto the global dynamic stiffness matrix of the final structure. The most important issue of the WW algorithm, J0 count, has been resolved with the analytical expressions derived for all possible cases. The proposed dynamic stiffness method (DSM) is then applied to several examples including individual membranes and their assemblies. High accuracy and exactness of the proposed method within the whole frequency range is demonstrated by comparison with the finite element method. Besides, interesting findings have been observed on repeated eigenvalues with distinct mode shapes corresponding to certain aspect ratio and tension ratio, where physical and mathematical understanding has been provided.

Dynamic modeling and characteristic analysis of the periodically coupled plate structure based on the dynamic stiffness method

Abstract In this paper, the dynamic model of periodically coupled plate structure is formulated by the dynamic stiffness method (DSM). Geometrical construction is assumed to vary periodically in the whole plate structure. According to the coupling condition, the whole periodic plate structure is divided into appropriate individual rectangular plates firstly. To the individual plate, the displacement fields both of in-plane and out-of-plane are derived from the governing equation with a strong-form methodology. The dynamic stiffness matrix is developed by the Projection method. Once the dynamic stiffness matrixes of the individual plates are obtained, the global dynamic stiffness matrix of the whole periodic structure can be assembled in the similar manner with the finite element method (FEM). Several examples are presented to validate the convergence and accuracy of the present formulation. From the harmonic responses, the band gap behaviors can be observed obviously. Furthermore, a parametric analysis is implemented to study the effects of geometrical distributions and plate dimensions on the vibration characteristics.

An exact modal analysis approach to vibration analysis of structures with mass-spring subsystems and rotational joints

A novel exact modal analysis approach is presented for vibration analysis of plane continuous structures, which are coupled with discrete mass-spring subsystems and include elastic rotational joints modelling local flexibility. Using the theory of generalised functions to handle the discontinuities of the response variables, every continuous member with any number of mass-spring subsystems and joints is treated as a two-node element, for which a 6 × 6 exact dynamic stiffness matrix is obtained in closed form. As a result, the global dynamic stiffness matrix is built by a standard finite-element assembling procedure, with size depending only on the number of nonzero nodal degrees of freedom of member-to-member nodes. Upon deriving pertinent orthogonality conditions for the modes, the system response under arbitrary loads is obtained by modal impulse and modal frequency response functions, under the assumption of proportional damping. The solutions are exact and can be used as benchmark for classical finite-element solutions. The approach is formulated for various mass-spring subsystems, acting in transverse and axial directions relative to every member.

Natural frequencies and harmonic responses of multi-story frames using single variable shear deformation theory

This study aims to perform free and forced vibration analyses of multi-story frames using the single variable shear deformation theory (SVSDT). The free vibration analysis begins with constructing dynamic stiffness matrices based on analytical solutions to motion equations for frame members according to SVSDT, as well as the Timoshenko beam theory. The natural frequencies are calculated via a global dynamic stiffness matrix of the frame for different column and beam dimensions, and elastic modulus values. The natural frequency results are tabulated with finite element method results to verify the dynamic stiffness method, thereby providing exact values. The mode shapes are plotted, and forced vibration analysis is accomplished using a global dynamic stiffness matrix of the frame for different dynamic lateral load amplitude values. The harmonic response curves are depicted in a logarithmic scale to observe the resonance and anti-resonance frequencies. The numerical examples demonstrate that the SVSDT can be used effectively for the dynamic analysis of frames.

Analysis on the dynamic characteristic of a tensioned double-beam system with a semi theoretical semi numerical method

In this study, the dynamic characteristic of an inclined and tensioned double-beam system is investigated. The double-beam system consists of two elastic beams, which are quite different in mass and stiffness, and are continuously connected by a layer of elastic springs. The beam with larger stiffness and mass is under a tensile axial loading. The oscillatory differential equations of this double-beam system are established by considering the effects of sag, flexural rigidity, boundary conditions, inclined angle of real inclined beams, and other factors simultaneously. Based on the governing equations, the element transverse dynamic stiffness matrix and global transverse dynamic stiffness matrix are derived to obtain the dynamic equilibrium equation of the system in a dynamic stiffness form. Using this, the system is simplified into a four degree-of-freedom simple oscillatory system and consequently the theoretical frequency characteristic equation is proposed for this double beam system. A numerical equation rooting approach is developed to solve the dynamical properties of the proposed equation. With the numerical case studies, the dynamic characteristics and its variation laws of a double-beam system are investigated. It shows that the proposed semi theoretical semi numerical methods can give an accurate solution for the double beam system, and rules revealed in this study are help for comprehending the dynamical behavior of double beam like engineering structures theoretically.