Dynamic Stiffness Formulation Research Articles

Effective dynamic properties of multi-segment beam lattices: a dynamic stiffness formulation

Effective dynamic properties of multi-segment beam lattices: a dynamic stiffness formulation

Dynamic stiffness formulation for vibration characteristics analysis of bi-dimensional functionally graded annular plate of variational thickness

Dynamic stiffness formulation for vibration characteristics analysis of bi-dimensional functionally graded annular plate of variational thickness

Dynamic stiffness formulations for exact modal and dynamic response analysis of three-dimensional acoustic cavities in cylindrical coordinates

Exact dynamic stiffness (DS) models for three-dimensional (3D) cavities in cylindrical coordinates under any classical acoustic boundary conditions (BCs) are proposed in this paper. The paper first derives the exact general solutions that satisfy the governing differential equations. Then, these general solutions are used to develop the DS matrices for 3D cavities in cylindrical coordinates by substituting them into the acoustic pressure and normal velocity BCs. Subsequently, the Wittrick–Williams algorithm, along with the analytical expressions of the j0 count for various shaped cavities, is applied to compute natural frequencies. Finally, reliable and efficient techniques for dynamic response solution are employed. The DSM provides benchmark solutions for various 3D cavities. The convergence, universality, and efficiency of the proposed method in modal and response analysis are illustrated by comparing the results obtained using the DSM and FEM. The calculation time when using the DSM is only 0.1% of that when using the commercial finite element method, while maintaining the same accuracy. This research not only provides an effective technique for analyzing the cylindrical acoustic pressure field in engineering, but also establishes a foundation for the analysis of structural–acoustic coupling in tubular structures.

Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels

In this paper, an exact dynamic stiffness formulation using higher-order theories with displacement variables only is presented and subsequently used to investigate the free vibration characteristics of solid beams, thin-walled structures and reinforced panel structures. In essence, higher-order displacement fields are developed by using the Carrera unified formulation (CUF), and by discretizing the cross-section kinematics with bilinear, cubic and fourth-order Lagrange polynomials. In particular, the component-wise (CW) approach based on Lagrange expansion is applied in which the solid part and thin-walled part are considered as two independent components that can be assembled. The principle of virtual displacements is used to derive the governing differential equations and the associated natural boundary conditions. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The explicit terms of the dynamic stiffness matrices are also presented. The Wittrick–Williams algorithm of the dynamic stiffness method (DSM) is applied with the explicit expressions of the J0 count for beam elements under all above support conditions. In return, there is no need to refine the element in the DSM, and thus, it becomes immensely efficient. The accuracy and efficiency of the proposed methodology are extensively assessed for different solid beams, thin-walled structures and reinforced panels and the results are compared with those appearing in published literature and also checked by 3D finite element (FE) solutions.

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.

Closed-form dynamic stiffness formulations for exact modal analysis of membranes in polar coordinates

Exact dynamic stiffness (DS) formulations for free vibration of membranes and their assemblies in polar coordinates are presented in this paper. First, the exact general solutions of the governing differential equations of membranes in polar coordinates are derived, in the form of Bessel functions. Then, by substituting the general solutions into the displacement and force boundary conditions (BCs), the frequency-dependent elemental DS matrices of various shaped membranes are developed. Finally, assembling the elemental stiffness matrices and applying BCs leads to the final DS matrix. By utilizing the Wittrick–Williams (W–W) algorithm, the natural frequencies and mode shapes are computed. The difficult problem in applying the W–W algorithm, the so called j0 count, is resolved by determining the zeros of Bessel equation and applying an indirect method. The proposed method is applied to representative examples of individual membranes and their assemblies under arbitrary BCs. The efficiency and accuracy of the dynamic stiffness method (DSM) are demonstrated by comparing results by the DSM with those by the finite element method (FEM). The proposed method can serve as an exact but efficient tool for modal analysis of membranes in structural dynamics as well as fluid–structure interaction problems.

Closed-form dynamic stiffness formulation for modal and dynamic response analysis of pile group foundations

A closed-form dynamic stiffness (DS) formulation is proposed for pile group foundations. This model consists of a rigid cap and an arbitrary number of flexible piles. The rigid cap, connecting to all piles, is modeled as a rigid body resting on an elastic foundation. For each pile, a longitudinal soil–pile model and a preloaded Timoshenko beam on viscoelastic Pasternak foundation are adopted to describe the longitudinal and transverse vibration, respectively. The piles can be either friction piles or end-bearing piles, and pile sections surrounded by different soil layers can be easily modeled by the assembly of pile elements with different parameters. The proposed DS formulations essentially use very few degrees of freedom to, nevertheless, describe the broadband dynamic behaviors of complex pile group foundations in a highly accurate manner. The DS formulation is then used for modal and dynamic response analysis. For modal analysis, the Wittrick–Williams algorithm is applied, with the key issue J0 count problems resolved. The proposed method is verified against some existing results. This research provides an exact and highly efficient modal and response analysis tool for pile group foundations, which can be used for the dynamic analysis of pile foundations subjected to earthquake or traffic excitations.

Dynamic stiffness formulation for free vibration analysis of rotating cross-ply laminated combined elliptical-cylindrical-conical shell

Dynamic stiffness formulation for free vibration analysis of rotating cross-ply laminated combined elliptical-cylindrical-conical shell

Dynamic stiffness formulation for vibration analysis of an open cylindrical shell and its coupling structures based on a generalized superposition method

• A dynamic stiffness formulation is provided for open cylindrical shells. • The dynamic stiffness formulation is derived by combining a generalized superposition method with the projection method. • Compared with available ones, the derivation process of the dynamic stiffness matrix is easier in the present formulation and its dimension is also largely reduced. • The present formulation is appropriate for various boundary conditions. • The present formulation is extended to the vibration analysis of coupled shell structures. • Parametric studies are performed on the vibration of coupled shell structures. This paper presents a dynamic stiffness formulation for free vibration analysis of open cylindrical shells and their coupling structures. Based on the Flügge's thin shell theory, the dynamic stiffness matrix of an open cylindrical shell is derived by employing the projection method and a newly developed superposition method, which considers the entire domain of the system as a whole part instead of dividing it into small subdomains in the traditional Gorman's superposition method. By doing this, the derivation process of the dynamic stiffness formulation is much easier and the analytical model has a significantly lower order than the traditional dynamic stiffness formulation. The presented dynamic stiffness formulation is then extended to the vibration analysis of coupled shell structures. To achieve this goal, the coupled shell structures are divided into several sub-shells, and the global dynamic stiffness matrices are assembled according to the geometrical coupled conditions between them. Several typical coupled shell structures are taken as examples and their vibration characteristics are studied by the present formulation. The convergence and accuracy of the present formulation are verified by comparing present results with those obtained by literature and the finite element method (FEM). Parameter studies for the vibration analysis of the coupled shell structures are presented, indicating that the circumference angles, length-to-radius ratios, and thickness-to-radius ratios have significant influences on the dynamic characteristics of the coupled shell structures.

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.

Dynamic amplification in masonry arch railway bridges

Dynamic amplification of loads in masonry arch railway bridges is not well understood. There is a scarcity of experimental data and previous numerical studies have only addressed a few specific bridge geometries. Despite this, guidance documents provide empirical and unvalidated formulae to calculate dynamic amplification for masonry arch railway bridges. To improve our fundamental understanding of the problem, determine appropriate modelling strategies and evaluate the reliability of guidance documents, simple 2D and 3D models are explored in this paper. The 2D approach idealises key bridge components (pier, arch, fill and backing elements) with straight Timoshenko beams, springs and lumped masses. It uses an analytical dynamic stiffness formulation, which is computationally efficient and well-suited to explore a range of bridge models. The higher fidelity 3D modelling approach uses shell and solid finite elements and is used to evaluate the limitations of 2D models. In both approaches, linear-elastic material behaviour is assumed and train loads are idealised as moving vertical loads distributed over an effective area. The modelling results indicate a complex relationship between train speed and dynamic amplification that depends critically on bridge geometry and axle spacing. In general, the multi-span bridge configurations experienced higher dynamic amplification over operational train speeds. The results also higlight deficiencies in existing code provisions and demonstrates how efficient numerical models may replace these provisions.

Vibration suppression by mistuning acoustic black hole dynamic vibration absorbers

This paper presents a modified add-on strategy to reduce the transverse vibration of beam structures using multiple acoustic black hole dynamic vibration absorbers (ABH-DVAs). The exact dynamic stiffness formulation of the system constituted by the host beam and arrays of ABH-DVAs is established. The intentional mistuning effect is introduced into the system by varying the taper lengths of ABH-DVAs, aiming at broadband vibration suppression. Several case studies are presented to demonstrate the effectiveness of the proposed strategy. In addition, the multi-objective optimization of the structure with minimum weight and broad attenuation band is investigated. From the numerical and experimental results, it is seen that the attenuation bandwidth of the proposed design is widened compared with that of the conventional design. The proposed strategy shows great potential for widening the attenuation band of the acoustic black hole structures.

Dynamic stiffness formulation for isotropic and orthotropic plates with point nodes

In this paper, the dynamic stiffness method for isotropic and orthotropic rectangular plates with point nodes is developed, making it possible to integrate the dynamic stiffness properties for plates with the dynamic stiffness properties of other elements such as bars and beams, but importantly, the advanced theory allows amalgamation of the dynamic stiffness method with the conventional finite element method for the first time. The derivation of the dynamic stiffness matrices for isotropic and orthotropic plates with point nodes has been accomplished by implementing the Fourier coefficients of the boundary values of the amplitudes of forces and displacements of the plate to form the force–displacement relationship at nodal points, including the corners. This innovative objective has been achieved by developing a new form of discrete Fourier transform technique for modified trigonometric functions. Using some carefully chosen illustrative examples, the convergence of results is ascertained by using different number of node points and their locations on the plate edges. The proposed theory has substantial advantages over conventional dynamic stiffness theories for plates, particularly when applying non-classical different boundary conditions on plate edges. The computed numerical results are discussed with significant conclusions drawn.

A highly accurate spectral dynamic stiffness method for efficient broadband modal and dynamic response analysis of membranes assemblies with arbitrary boundary conditions

A highly accurate and efficient spectral dynamic stiffness (SDS) formulation is presented for the broadband dynamic analysis of damped membranes and their assemblies with arbitrary boundary conditions (BCs). First, the general solution satisfying exactly the governing differential equations is derived, and any general BCs are represented by the modified Fourier series (MFS). Second, the modified Fourier coefficients of the force BCs are associated with those of the displacement BCs by eliminating the coefficients in the general solution, where the SDS matrix is formulated. The SDS elements can be assembled through line nodes to model complex geometries and arbitrary BCs can be described. Finally, highly reliable and efficient eigenvalue and dynamic response solution techniques are employed. The performance of the proposed method is illustrated by convergence and efficiency studies. Modal analysis and damped dynamic response analysis are performed for either individual membranes or complex membrane assemblies subjected to general BCs. This method is demonstrated to be more versatile than the classical dynamic stiffness method, and yields highly accurate results with reasonable computational efficiency by comparison with the FEM. Due to these excellent features, this method can serve as a powerful alternative tool for the fluid–structure interaction and broadband vibro-acoustic problems.

Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes

This paper extends the Wittrick-Williams (W-W) algorithm for hybrid dynamic stiffness (DS) models connecting any combinations of line and point nodes. The principal novelties lie in the development of both the DS formulation and the solution technique in a sufficiently systematic and general manner. The parent structure is considered to be in the form of two dimensional DS elements with line nodes, which can be connected to rigid/spring point supports/connections, rod/beam point supports/connections, and point connections to substructures. This is achieved by proposing a direct constrain method in a strong form which makes the modeling process straightforward. For the solution technique, the W-W algorithm is extended for all of the above hybrid DS models. No matrix inversion is needed in the proposed extension, making the algorithm numerically stable, especially for complex built-up structures. A mathematical proof is provided for the extended W-W algorithm. The proposed DS formulation and the extended W-W algorithm are validated by the FE results computed by ANSYS. This work significantly extends the application scope of the DS formulation and the W-W algorithm in a methodical and reliable manner, providing a powerful eigenvalue analysis tool for beam-plate built-up structures.

Dynamic stiffness method for exact longitudinal free vibration of rods and trusses using simple and advanced theories

Closed-form dynamic stiffness (DS) formulations coupled with an efficient eigen-solution technique are proposed for exact longitudinal free vibration analyses of rods and trusses by using classical, Rayleigh-Love, Rayleigh-Bishop and Mindlin–Hermann theories. First, the exact general solutions of the governing differential equations of the four rod theories are developed. Then the solutions are substituted into the generalized displacement and force boundary conditions (BCs), leading to the elemental DS matrices utilising symbolic computation. As an accurate and efficient modal solution technique, the Wittrick-Williams (WW) algorithm is applied. The J0 count for the WW algorithm has been resolved for all four types of DS elements with explicit analytical expressions. The method is verified against some existing exact results for rods subjected to specific BCs. Comparisons of the natural frequencies and mode shapes for different theories and slenderness ratios are also made. Finally, benchmark solutions are provided for individual rods subject to different BCs, a stepped rod and a truss. This research provides an exact and highly efficient modal analysis tool for rods and trusses within the whole frequency range, which is suitable for parametric studies, optimization design, inverse problem analysis, and important for statistical energy analysis.

Closed-form dynamic stiffness formulation for exact modal analysis of tapered and functionally graded beams and their assemblies

The paper proposes a closed-form dynamic stiffness (DS) formulation for exact transverse free vibration analysis of tapered and/or functionally graded beams based on Euler–Bernoulli theory. The novelties lie in both the DS formulation and the solution technique. For the formulation, the developed DS is applicable to a wide range of non-uniform beams whose bending stiffness and linear density are assumed to be polynomial functions of position. This fills a gap of existing closed-form DS element library which is generally limited to linearly tapered/functionally graded beams. For the solution technique, an elegant and efficient J0 count of tapered element is proposed to apply the Wittrick-Williams (WW) algorithm most effectively. The investigation sheds lights on the so-called J0 count challenge of the algorithm for other DS elements. The above two novelties make exact and highly efficient modal analysis possible for a wide range of tapered and/or functionally graded beams, without resorting to series solution, numerical integrations or refined mesh discretization. Results for a particular case show excellent agreement with published results. Moreover, we investigate the effects of the taper/functional gradient rate/index and boundary conditions on the free vibration behaviour. Benchmark solutions are provided for individual beams as well as beam assemblies.

Earthquake response of linear-elastic arch-frames using exact curved beam formulations

PurposeThis study aims to obtain earthquake responses of linear-elastic multi-span arch-frames by using exact curved beam formulations. For this purpose, the dynamic stiffness method (DSM) which uses exact mode shapes is applied to a three-span arch-frame considering axial extensibility, shear deformation and rotational inertia for both columns and curved beams. Using exact free vibration properties obtained from the DSM approach, the arch-frame model is simplified into an equivalent single degree of freedom (SDOF) system to perform earthquake response analysis.Design/methodology/approachThe dynamic stiffness formulations of curved beams for free vibrations are validated by using the experimental data in the literature. The free vibrations of the arch-frame model are investigated for various span lengths, opening angle and column dimensions to observe their effects on the dynamic behaviour. The calculated natural frequencies via the DSM are presented in comparison with the results of the finite element method (FEM). The mode shapes are presented. The earthquake responses are calculated from the modal equation by using Runge-Kutta algorithm.FindingsThe displacement, base shear, acceleration and internal force time-histories that are obtained from the proposed approach are compared to the results of the finite element approach where a very good agreement is observed. For various span length, opening angle and column dimension values, the displacement and base shear time-histories of the arch-frame are presented. The results show that the proposed approach can be used as an effective tool to calculate earthquake responses of frame structures having curved beam elements.Originality/valueThe earthquake response of arch-frames consisting of curved beams and straight columns using exact formulations is obtained for the first time according to the best of the author’s knowledge. The DSM, which uses exact mode shapes and provides accurate free vibration analysis results considering each structural members as one element, is applied. The complicated structural system is simplified into an equivalent SDOF system using exact mode shapes obtained from the DSM and earthquake responses are calculated by solving the modal equation. The proposed approach is an important alternative to classical FEM for earthquake response analysis of frame structures having curved members.

Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

Extended Wittrick–Williams algorithm for eigenvalue solution of stochastic dynamic stiffness method

This paper proposes an efficient and reliable eigenvalue solution technique for analytical stochastic dynamic stiffness (SDS) formulations of beam built-up structures with parametric uncertainties. The SDS formulations are developed based on frequency-dependent shape functions in conjunction with both random-variable and random-field structural parameters. The overall numerical framework is aimed towards representing the broadband dynamics of structures using very few degrees of freedom. This paper proposes a novel approach combining the Wittrick–Williams(WW) algorithm, the Newton iteration method and numerical perturbation method to extract eigensolutions from SDS formulations. First, the eigenvalues and eigenvectors of the deterministic DS formulations are computed by the WW algorithm and the corresponding mode finding technique, which are used as the initial solution. Then, a numerical perturbation technique based on the inverse iteration and homotopy method is proposed to update the eigenvectors and eigenvalues. The robustness and efficiency of the proposed method are guaranteed through several technique arrangements. Through numerical examples, the proposed method is demonstrated to be robust within the whole frequency range. This method provides an efficient and reliable tool for stochastic analysis of eigenvalue problems relevant to free vibration and buckling analysis of built-up structures.