Dynamic Stiffness Elements Research Articles

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.

Stochastic dynamic stiffness for damped taut membranes

An analytical stochastic dynamic stiffness formulation is developed for the dynamic analysis of damped membrane structures with parametric uncertainties. First, the exact general solution of a biaxially taut membrane in the frequency domain is derived, which is used as the frequency-dependent shape function. Both the material properties and the tension fields of the membrane are modelled as 2D random fields with an exponential autocorrelation function in both x and y directions. Then, the random fields are decomposed by Karhunen-Loève (KL) expansion. After a formulation procedure like the finite element method, the stochastic stiffness, and mass elemental matrices are derived based on the frequency-dependent shape function and the KL expansion, subsequently forming the stochastic dynamic stiffness matrix. The developed stochastic dynamic stiffness elements can be assembled to model membrane assemblies with general boundary conditions considering uncertainties. The proposed method can be utilized as a feasible technique for the efficient and accurate stochastic dynamic analysis in the whole frequency domain. The current research paves the way for stochastic dynamic stiffness formulation for other two-dimensional structures like plates and shells.

Application of operational method to develop dynamic stiffness matrix for vibration analysis of thin beams

Laplace transform operator only accepts initial values at zero in evaluating derivative functions. This limits its use in vibration analysis of beams as a tool for merely finding analytical solutions and coefficients. By extending Laplace transform’s capacity to take non-zero initial conditions, a novel operational method for developing dynamic beam elements is proposed. The method avoids integration and differentiation, which are traditionally used in developing dynamic vibration elements. The ease of application of this operational method is demonstrated by developing dynamic stiffness elements of Euler-Bernoulli beams under various boundary conditions. The proposed technique also allows for easy handling of coupled differential equations, such as the ones applied to bending-torsion vibration of beams, as well as discontinuous functions which appear in elastic support modelling. MATLAB® software has been used as graphing tool and for the mathematical calculations in this research.

Analysis of Vibration of the Euler-Bernoulli Pipe Conveying Fluid by Dynamic Stiffness Method and Transfer Matrix

The dynamic stiffness method and Transfer method is applied to study the vibration characteristics of the Euler-Bernoulli pipe conveying fluid in this paper. According to the dynamics equation of the pipe conveying fluid, the element dynamic stiffness is established. The vibration characteristic of the single-span pipe is analyzed under two kinds of boundary conditions. The results compared with the literature, which has a good consistency. Based on this method, natural frequency and the critical speed of the two types of multi-span pipe are deserved. This paper shows that the dynamic stiffness method and transfer matrix is an effective method to deal with the vibration problem of pipe conveying fluid.

Framework for Dynamic‐Stiffness‐Based Free Vibration Analysis of Plate‐Like Structures

A framework for free vibration analysis of plate‐like structures is presented in the paper. Based on the previously formulated dynamic stiffness elements, FREEVIB object‐oriented software in Python environment has been created. Software design and structure as well as a wide range of possible structural problems that could be analyzed using the FREEVIB are presented. Through several illustrative examples including free vibration analysis of stepped, stiffened and folded plate structures, implying isotropic or orthotropic material formulations, the efficiency and accuracy of the FREEVIB is demonstrated. The possibilities of further extensions and improvements of the software are discussed.

A new prestressed dynamic stiffness element for vibration analysis of thick circular cylindrical shells

This paper presents a new spectral element based on the dynamic stiffness matrix of a prestressed cylindrical shell. The dynamic stiffness matrix is built using first-order shear deformation theory, and natural frequencies are processed easily. Vibration analyses are performed with numerical examples to determine the performance of this approach and the effect of prestressing on frequency response functions. This element has many advantages over the finite element method in terms of accuracy, model size and computing time.

Free vibration of functionally graded beams and frameworks using the dynamic stiffness method

The free vibration analysis of functionally graded beams (FGBs) and frameworks containing FGBs is carried out by applying the dynamic stiffness method and deriving the elements of the dynamic stiffness matrix in explicit algebraic form. The usually adopted rule that the material properties of the FGB vary continuously through the thickness according to a power law forms the fundamental basis of the governing differential equations of motion in free vibration. The differential equations are solved in closed analytical form when the free vibratory motion is harmonic. The dynamic stiffness matrix is then formulated by relating the amplitudes of forces to those of the displacements at the two ends of the beam. Next, the explicit algebraic expressions for the dynamic stiffness elements are derived with the help of symbolic computation. Finally the Wittrick-Williams algorithm is applied as solution technique to solve the free vibration problems of FGBs with uniform cross-section, stepped FGBs and frameworks consisting of FGBs. Some numerical results are validated against published results, but in the absence of published results for frameworks containing FGBs, consistency checks on the reliability of results are performed. The paper closes with discussion of results and conclusions.

Free vibration analysis of stiffened and cracked laminated composite plate assemblies using shear-deformable dynamic stiffness elements

In the paper, the dynamic stiffness method based on the HSDT and FSDT plate theories is applied to study free vibration characteristics of composite stiffened and cracked plate assemblies. The dynamic stiffness matrices for transverse vibration of laminated composite plate based on the HSDT and FSDT previously derived by authors have been coupled with the dynamic stiffness matrix of laminated composite plate undergoing in-plane vibration. The numerical analysis has been carried out through several illustrative examples in order to check the applicability and accuracy of the proposed method in the free vibration analysis of stiffened plate assemblies. The effects of transverse shear deformation, boundary conditions, side-to-thickness ratios, crack length or reinforcement amount within the GFRP beam on the free vibration characteristics have been discussed, and the variety of new results has been provided as a benchmark for future investigations.

Application of the dynamic stiffness method in the vibration analysis of stiffened composite plates

Composite laminates are nowadays extensively applied in many engineering disciplines. Free vibration characteristics of such structures are not always easy to predict by using conventional finite element method (FEM). As an alternative, the dynamic stiffness method (DSM) can be applied to predict free vibration characteristics of composite plate assemblies, especially in mid and high frequency ranges. Key feature of the DSM is the dynamic stiffness element (DSE) and its dynamic stiffness matrix, derived from the exact solution of the governing equations of motion in the frequency domain. Consequently, the structural discretization is influenced only by the change in the geometrical and/or material properties of the structure. The number of unknowns is significantly decreased in comparison with the FEM, without losing the accuracy and reliability of the results.In the paper, the DSE based on the higher order shear deformation theory (HSDT) is applied to study free vibration analysis of composite stiffened plates. The numerical analysis has been carried out through an illustrative example in order to check the accuracy of the proposed method. The influence of side-to-thickness ratio on the free vibration characteristics of stiffened plate has been studied numerically. The results are validated using the available analytical data as well as with the FEM solutions.

Primena metode dinamičke krutosti u numeričkoj analizi slobodnih vibracija ploča sa ukrućenjima

The free vibration analysis of stiffened plate assemblies has been performed in this paper by using the dynamic stiffness method. Rectangular Mindlin plate dynamic stiffness element has been formulated. Using the rotation matrices, dynamic stiffness matrices of single plates have been derived in global coordinate system. The global dynamic stiffness matrix of plate assembly has been derived by using similar assembly procedure as in the finite element method. The natural frequencies of stiffened plate assemblies with different boundary conditions have been computed and validated against the results obtained by using the commercial software package Abaqus. High accuracy of the results has been demonstrated.

Multimodal vibration damping of a plate by piezoelectric coupling to its analogous electrical network

Multimodal damping can be achieved by coupling a mechanical structure to an electrical network exhibiting similar modal properties. Focusing on a plate, a new topology for such an electrical analogue is found from a finite difference approximation of the Kirchhoff–Love theory and the use of the direct electromechanical analogy. Discrete models based on element dynamic stiffness matrices are proposed to simulate square plate unit cells coupled to their electrical analogues through two-dimensional piezoelectric transducers. A setup made of a clamped plate covered with an array of piezoelectric patches is built in order to validate the control strategy and the numerical models. The analogous electrical network is implemented with passive components as inductors, transformers and the inherent capacitance of the piezoelectric patches. The effect of the piezoelectric coupling on the dynamics of the clamped plate is significant as it creates the equivalent of a multimodal tuned mass damping. An adequate tuning of the network then yields a broadband vibration reduction. In the end, the use of an analogous electrical network appears as an efficient solution for the multimodal control of a plate.

Shear deformable dynamic stiffness elements for a free vibration analysis of composite plate assemblies – Part II: Numerical examples

The dynamic stiffness elements based on the HSDT and FSDT (and the corresponding dynamic stiffness matrices of a laminated composite plate) have been implemented into the original MATLAB code, and a variety of numerical investigations is performed in this paper. The obtained results are validated against the exact and numerical solutions or the experimental data from the literature. In the absence of the analytical solutions, the numerical solutions obtained by using the Abaqus software are used. Excellent agreement is achieved against the existing exact solutions.After the detail convergence study considering both low and high modes of vibration, the effects of the plate side-to-thickness and orthotropy ratios, as well as the influence of boundary conditions, have been discussed by comparing the proposed models with the exact Levy-type solutions from the literature. A variety of new results for non-special boundary conditions is provided. Two additional examples of a completely free laminated composite plate undergoing free vibrations are provided and the results are validated against the experiment. Finally, the analyses considering the Z- and U-shaped composite plate assemblies with various boundary conditions are provided as a benchmark for future research.

Free vibration study of sandwich plates using a family of novel shear deformable dynamic stiffness elements: limitations and comparison with the finite element solutions

In this paper, the dynamic stiffness matrix of a completely free rectangular multi-layer plate element based on Reddy's higher-order shear deformation theory is derived. The reduction of the proposed model to the first-order shear deformation theory-based formulation is presented. Three coupled Euler-Lagrange equations of motion have been transformed into two uncoupled equations introducing a boundary layer function. The proposed model enables free transverse vibration analysis of rectangular multi-layer plates with (transversely) isotropic layers having arbitrary combinations of boundary conditions.The influence of transverse shear deformation is discussed along with the applicability of two shear deformable dynamic stiffness elements. Moreover, the influence of the boundary conditions on the free vibration characteristics of sandwich panels has been discussed. The natural frequencies obtained using different dynamic stiffness multi-layer plate elements have been validated against the solutions from the commercial software Abaqus and the previously verified numerical solutions using layered finite elements. The limitations of the model regarding the differences between material properties of the face and core layers within a sandwich plate are highlighted. The influence of face-to-core thickness ratio on natural frequencies is illustrated, while a variety of new results is provided as a benchmark for future investigations.

A Dynamic Finite Element for Coupled Extensional-Torsional Vibration of Uniform Composite Thin-Walled Beams

A Dynamic Finite Element (DFE) formulation for the free vibration analysis of extension-torsion coupled uniform composite thin-walled beams is presented. Employing the exact solutions of the differential equations governing the uncoupled vibrations of a uniform beam element, the analytical expressions for extensional and torsional dynamic trigonometric shape functions are derived. By exploiting the principle of virtual work and the frequency-dependent shape functions, the element dynamic stiffness matrix is developed. The application of the theory is demonstrated by a Circumferentially Uniform Stiffness (CUS) composite circular tube for which the influence of ply fibre-angle on the natural frequencies is studied. A variety of CUS configurations are studied and the correctness of the theory and the superiority of the proposed DFE over the conventional FEM methods are confirmed by numerical checks and the published results.

Modal analysis of sailplane and transport aircraft wings using the dynamic stiffness method

The purpose of this paper is to provide theory, results, discussion and conclusions arising from an in-depth investigation on the modal behaviour of high aspect ratio aircraft wings. The illustrative examples chosen are representative of sailplane and transport airliner wings. To achieve this objective, the dynamic stiffness method of modal analysis is used. The wing is represented by a series of dynamic stiffness elements of bending-torsion coupled beams which are assembled to form the overall dynamic stiffness matrix of the complete wing. With cantilever boundary condition applied at the root, the eigenvalue problem is formulated and finally solved with the help of the Wittrick-Williams algorithm to yield the eigenvalues and eigenmodes which are essentially the natural frequencies and mode shapes of the wing. Results for wings of two sailplanes and four transport aircraft are discussed and finally some conclusions are drawn

Detection of damage through coupled axial–flexural wave interactions in a sagged rod using the spectral finite element method

This paper presents the results of a computational and experimental validation exercise performed towards damage identification of a sagged rod with known damage by using the coupled axial–flexural wave interaction mechanics. Towards simulating the damage scenario in a sagged conductor made of steel wire rope, a prismatic steel rod is taken up for study. An initial axial wave, tangential to the curve of the arc, manifests as both axial and flexural waves as it propagates alongside the length of the rod. This interaction effect between axial and flexure wave propagation is studied in this paper. Impedance mismatch is made in the rod by changing its cross-sectional area along its length. Numerical simulations are implemented using the spectral finite element method with a combined axial and flexure effect. The concept of obtaining the exact spectral element dynamic stiffness matrix for a wave propagation analysis sagged rod is discussed. Computation is implemented in the Fourier domain using Fast Fourier Transform (FFT). In the time domain, post processing of the response is done, which is applicable in structural diagnostics in addition to the wave propagation problem. The predominant single-frequency-based amplitude-modulated, narrow-banded, burst wave propagation is found to be better matched if the elemental rod theory is replaced with a modified rod theory called the Love theory. The differences in the propagating waves allow identification of the damage location in a very clear-cut way. The methodology of the moving correlation coefficient is also successfully employed to detect the damage precisely. This fact is very encouraging for future work on structural health monitoring.

Free vibration analysis of composite plates by higher-order 1D dynamic stiffness elements and experiments

A novel approach for free vibration analysis of composite plate-like structures is introduced in this paper. Refined beam theories are formulated by making use of the Carrera Unified Formulation (CUF). By exploiting the hierarchical characteristics of CUF, the differential equations of motions and the natural boundary conditions are written in a compact and concise form in terms of fundamental nuclei, whose formal mathematical expressions do not depend on the order of the theory N. After the closed form solution of the N-th order beam model is sought, a general procedure to derive the exact Dynamic Stiffness (DS) matrix is devised by relating the amplitudes of the harmonically varying loads to those of the responses. The global DS matrices of composite laminated plates are then used with reference to the algorithm of Wittrick and Williams to carry out free vibration analyses. The accuracy of the proposed methodology is verified through published literature, finite element solutions from the commercial code MSC/Nastran and experimental tests.

Flutter analysis by refined 1D dynamic stiffness elements and doublet lattice method

An advanced model for the linear flutter analysis is introduced in this paper. Higher-order beam structural models are developed by using the Carrera Unified Formulation, which allows for the straightforward implementation of arbitrarily rich displacement fields without the need of a-priori kinematic assumptions. The strong form of the principle of virtual displacements is used to obtain the equations of motion and the natural boundary conditions for beams in free vibration. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The resulting dynamic stiffness matrix is used with particular reference to the Wittrick-Williams algorithm to carry out free vibration analyses. According to the doublet lattice method, the natural mode shapes are subsequently used as generalized motions for the generation of the unsteady aerodynamic generalized forces. Finally, the g-method is used to conduct flutter analyses of both isotropic and laminated composite lifting surfaces. The obtained results perfectly match those from 1D and 2D finite elements and those from experimental analyses. It can be stated that refined beam models are compulsory to deal with the flutter analysis of wing models whereas classical and lower-order models (up to the second-order) are not able to detect those flutter conditions that are characterized by bending-torsion couplings.