Dynamic Stiffness Matrix Research Articles

Delamination effects on the unsteady aero-elastic behavior of composite wing by modal analysis

This article deals with the investigation of delamination effects on the flutter behavior of a composite wing subjected to unsteady aerodynamic loads by providing an integrated procedure using exact methods. For this purpose, an undamaged wing structure is modeled as a cantilevered Bernoulli–Euler beam with coupled bending–torsion and delaminated wing structure is modeled as three interconnected intact beams. The modal analysis of the intact and delaminated beams is performed and mode shapes are obtained based on the development of the dynamic stiffness matrix, a computer code and an iterative method. The fundamental modes of the intact and delaminated beam derived from the free vibration analysis are used in Galerkin’s method and the flutter and divergence speeds are obtained with aerodynamic forces applied. In fact, modified shape functions were developed to reflect the delamination effect in the aeroelastic analysis. Also, for the unsteady aeroelastic analysis, an iterative algorithm is developed. It was found that the delamination location and its length can have a positive or a negative influence on the flutter behavior of the wing which depends on the fiber angle.

Out-of-plane free vibration analysis of three-layer sandwich beams using dynamic stiffness matrix

This paper has been devoted to exact calculation of the natural frequencies for out-of-plane free vibration of three-layer symmetric sandwich beams by using dynamic stiffness matrix method. Governing differential equations of motion have been derived using Euler-Bernoulli beam theory and St. Venant’s torsion theory, so that the effects of shear deformation in the face layers and rotary inertia are not taken into account and the cross-section is allowed to warp without restraint. Wittrick-Williams algorithm has been used to calculate the natural frequencies. Accuracy and ability of the dynamic stiffness matrix method have been illustrated in form of several examples. Results are verified through comparison with the natural frequencies obtained from ABAQUS software. Full independence of out-of-plane bending and torsional modes of vibration for three-layer symmetric sandwich beams has been proven.

Forced vibration analysis of viscoelastic helical rods with varying cross-section and functionally graded material

In scope of this study, forced vibration analysis of viscoelastic helical rods with varying cross-section and functionally graded material are investigated. Differential equations governing the dynamic behavior of helical rods are obtained in Laplace domain by the Timoshenko’s beam theory. Material and section geometry are assumed to be varying functionally along the rod axis. Viscoelasticity of the material is implemented via Kelvin’s model. Stiffness and transfer matrix methods are used together in order to obtain dynamic stiffness matrix of the system. Acquired results in Laplace domain are converted to time domain by using Durbin’s inverse Laplace algorithm. A parametric study is carried out for the investigation of the effects of material variation, non-uniformity and damping on the forced vibration of functionally graded viscoelastic rods.

Flexural Waves in Elastically Coupled Telescopic Metabeams

Abstract This paper investigates the flexural wave propagation through elastically coupled telescopic metabeams. It is assumed that the metabeam is formed by connecting successive beams with each other using distributed elastic springs. The equations of motion of a representative unit of the above-mentioned novel structural form are established by dividing it into three constitutive components that are two side beams, modeled employing the Euler–Bernoulli beam equation and an elastically coupled articulated distributed spring connection (ECADSC) at middle. ECADSC is modeled as parallel double beams connected by distributed springs. The underlying mechanics of this system in context of elastic wave propagation is unique when compared with the existing state of art in which local resonators, inertial amplifiers, etc. are attached to the beam to widen the attenuation bandwidth. The dynamic stiffness matrix is employed in conjunction with Bloch–Floquet theorem to derive the band structure of the system. It is identified that the coupling coefficient of the distributed spring layer and length ratio between the side beams and the elastic coupling plays the key role in the wave attenuation. It has been perceived that a considerable widening of the attenuation bandgap in the low frequency can be achieved while the elastically distributed springs are weak and distributed in a small stretch. Specifically, 140% normalized bandgap can be obtained only by tuning the stiffness and the length ratio without adding any added masses or resonators to the structure.

Continuum-based approach for modelling the flexural behaviour of plain concrete beam under high-cycle fatigue loads

In this study, a continuum-based model was proposed to characterise the flexural performance and damage evolution process of a plain concrete beam under high-cycle fatigue loads. The plain concrete beam under three-point bending load was modelled by combining the damaged constitutive model of concrete and the Euler–Bernoulli beam theory with varied stiffness. The dynamic stiffness matrix method, in conjunction with the discretization technique, was adopted to solve the nonlinear governing equations of motion. A global damage index corresponding to natural frequencies was introduced to quantify the performance degradation of the beam. To improve the analysis efficiency of the high-cycle fatigue problems, the accelerated algorithm, namely the jump-in-cycle method, was employed in this model. It is demonstrated that the numerical results agree well with the experimental data under fatigue harmonic loads. Performance degradation under fatigue bending loads only occurs at the midspan of the beam. Adopting the jump-in-cycle methods significantly improves the analysis efficiency of the high-cycle fatigue problem, and the calculation time is reduced by approximately 90% compared with the cycle-by-cycle method. This model is capable of rationally predicting damage evolutions, stiffness degradation processes, stress redistributions, and loading level-fatigue life (S–N) curves and can provide a basis for simulating the flexural behaviours of reinforced concrete (RC) beams under fatigue bending loads.

Unsteady aero-elastic analysis of a composite wing containing an edge crack

This paper deals with the investigation of the influence of edge surface crack on the flutter behavior of unidirectional fiber-reinforced composite wing subjected to unsteady aerodynamic loads. The cracked wing is modeled as a two-interconnected Euler–Bernoulli beam at the crack location, where the related crack is modeled using continuity conditions and the local flexibility matrix concept. The components of this matrix are extracted from the theory of linear fracture mechanics. The formulation takes into account the effects of bending-torsion coupling due to the unbalanced laminates and offset of the center of gravity inherent to composite structures. The fundamental modes of intact and cracked cantilevered beams, derived from the free vibration analysis by the dynamic stiffness matrix, are used in Galerkin's method. The modified shape functions are developed to reflect the crack effect in the aeroelastic analysis, wherein an unsteady aerodynamic model based on strip theory is employed. By a combination of structure and aerodynamic formulations and using the modified mode shapes and applying a new iteration algorithm, the flutter boundary is determined. Also, the procedure has been evaluated by solving the number of problems available in the literature. Then effects of various parameters, such as the fiber angle, the crack location, the crack length, and the aerodynamic model, on the flutter boundary are analyzed and a detailed discussion will be addressed. It was found that these parameters can have a positive or negative influence on the flutter behavior of the structure.

Damage detection in nonlinear vibrating structures using model updating

This paper presents two distinct model updating strategies for dynamical systems with local nonlinearities based on acceleration time history responses measured spatially across the vibrating structure. Both linear and nonlinear parameters are calibrated by minimizing the selected metric based on measured and predicted response using the newly proposed variant of differential search algorithm named as multi-cluster hybrid adaptive differential search (MCHADS) algorithm. The first model updating strategy involves the decoupling of linear and nonlinear characteristics of the system. In this scheme, we first establish the dynamic stiffness matrix using input and output measurements and then the underlying linear system is alone updated using it. Later, localization of the nonlinear attachment is attempted using the inverse property of the FRF of the nonlinear system and the established dynamic stiffness matrix of the underlying linear system from the previous step. Once the linear system is updated and with the already identified location(s) of nonlinear attachment(s), the nonlinear parameters are identified by formulating it as an optimization problem using the proposed MCHADS algorithm. The major advantage of the first approach is the reduction of the complex problem of nonlinear model updating to linear model updating. The second approach involves updating both linear and nonlinear parameters simultaneously using the proposed MCHADS algorithm. Investigations have been carried out by solving several numerically simulated examples and also with the experimental data of a benchmark problem to evaluate the effectiveness of the two proposed nonlinear model updating strategies. Further, investigations are also carried out to evaluate the capability of the model updating approaches towards damage identification in initially healthy nonlinear systems. Conclusions are drawn based on these investigations, highlighting the strengths and weaknesses of the two model updating approaches

Simply structured controllers for vibration suppression in long rotors

In this paper suppression of the transient flexural vibrational disturbances in long rotors, with fluid film bearings, is investigated. The rotor is described by a series of distributed shafts connected by the lumped discs, and the system is mounted on lumped fluid film bearings. Upon determination of the dynamic stiffness matrix of the system, the best approximate transfer function matrix description of the rotor, is determined. Initially vibration suppression by simple diagonal Proportional + Integral (PI) controllers is studied and via direct search optimisation techniques the PI parameters which exhibit fast vibration suppression is found. The resulted high integration rate, and low proportional gain PI controller, theoretically provided fast suppression time. However, it is shown that due to the strong coupling effect in the rotor system, and high rate of integration, the closed loop relative stability is weak, and feasibility of controller is questionable. Therefore, an alternative simple first order controller without integration action, that is named “attenuation filter “is suggested that can produce stronger stability and produces significant (not full) vibration suppression. The closed loop multivariable control of the rotor system comprising two vibration sensors and two magnetic actuators using such attenuation filter, is then simulated. The response to step disturbances, has provided 95% suppression with significantly fast response. It is concluded that although the attenuation filter may not provide 100% suppression, but it more reliable since the integration of the error, that results weak stability is avoided.

REALIZATION OF GROUND MOTION INPUT IN ABAQUS FOR LAYERED FOUNDATION UNDER SV WAVE OF OBLIQUE INCIDENCE OVER CRITICAL ANGLE

In the finite element numerical simulation, it is an urgent challenge to determine the ground motion input for the layered foundation under SV wave of oblique incidence over critical angle. Exact dynamic stiffness matrix in the frequency domain (i.e., frequency domain stiffness matrix method) of layered foundation is used to derive the formula for calculating the equivalent node force of ground motion input under arbitrary angular oblique incidence of SV wave, and the validity and accuracy of the ground motion input applied to ABAQUS numerical simulation are demonstrated by simulating seismic wave field of homogeneous half-space and layered foundation under oblique incidence of SV wave through ABAQUS software. It is shown that the inclined SV wave incidence with an arbitrary angle can be realized through ABAQUS simulation by means of stiffness matrix method, and the method is of high accuracy, especially in the case of SV wave incidence with an inclined angle greater than the critical one. The oval like motion trails of surface particles in uniform half space and the wave motion characteristics of layered half space are simulated commendably by finite element numerical modelling. On the above basis, the frequency domain stiffness matrix method is further combined with equivalent linearization method, which solves the ground motion input problem of two-dimensional layered half space considering soil nonlinearity for the inclined incidence with arbitrary angle.

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.

Exact solution by dynamic stiffness method for the natural vibration of porous functionally graded plate considering neutral surface

This paper presents the formulation of dynamic stiffness matrix for the natural vibration analysis of porous power-law functionally graded Levy-type plate. In the process of formulating the dynamic stiffness matrix, Kirchhoff-Love plate theory in tandem with the notion of neutral surface has been taken on board. The developed dynamic stiffness matrix, a transcendental function of frequency, has been solved through the Wittrick–Williams algorithm. Hamilton’s principle is used to obtain the equation of motion and associated natural boundary conditions of porous power-law functionally graded plate. The variation across the thickness of the functionally graded plate’s material properties follows the power-law function. During the fabrication process, the microvoids and pores develop in functionally graded material plates. Three types of porosity distributions are considered in this article: even, uneven, and logarithmic. The eigenvalues computed by the dynamic stiffness matrix using Wittrick–Williams algorithm for isotropic, power-law functionally graded, and porous power-law functionally graded plate are juxtaposed with previously referred results, and good agreement is found. The significance of various parameters of plate vis-à-vis aspect ratio ( L/b), boundary conditions, volume fraction index ( p), porosity parameter ( e), and porosity distribution on the eigenvalues of the porous power-law functionally graded plate is examined. The effect of material density ratio and Young’s modulus ratio on the natural vibration of porous power-law functionally graded plate is also explained in this article. The results also prove that the method provided in the present work is highly accurate and computationally efficient and could be confidently used as a reference for further study of porous functionally graded material plate.

Flexural edge waves in a Kirchhoff plate carrying periodic edge resonators and resting on a Winkler foundation

Flexural edge waves in a Kirchhoff plate, carrying periodically spaced spring–mass​ resonators at its edge and laid on a Winkler foundation, are considered. A spectral element method is applied on a representative unit cell to develop the dynamic stiffness matrix of the structure. In light of structural periodicity, Bloch wave theorem is used to derive a quadratic eigenvalue problem of the wave propagation constants, which are numerically solved for. Frequency bands are analyzed using the attenuation and phase constants. Two types of bandgaps are identified: one is attributed to the existence of attached resonators and the other is due to the fact that these resonators are arranged periodically. It is found that these bandgaps are influenced by the value of the resonator’s natural frequency and the stiffness of the elastic foundation. The analytically realized bandgap structures agree very well with the finite-element obtained dispersion curves by COMSOL.

Dynamic Stiffness Formulation for Out-of-Plane Natural Vibration of Elastically Supported Functionally Graded Plates

In this paper, the natural vibration characteristics of elastically supported functionally graded material plate are investigated using the dynamic stiffness method (DSM). Power-law functionally graded (P-FG) plate, the material properties of which vary smoothly along the thickness direction following the power-law function, that has been used for the analysis. Classical plate theory and Hamilton’s principle are used for deriving the governing differential equation of motion and associated edge conditions for P-FG plate supported by elastic foundation. During the formulation of dynamic stiffness (DS) matrix, the concepts of rotary inertia and neutral surface are implemented. Wittrick–Williams (W-W) algorithm is used as a solving technique for the DS matrix to compute eigenvalues. The results thus obtained by DSM for the isotropic, P-FG plate, and the P-FG plate with elastic foundation compare well with published results that are based on different analytical and numerical methods. The comparisons indicate that this approach is very accurate. Furthermore, results are provided for elastically supported P-FG plate under four different considerations in order to see the differences in frequencies with the inclusion or exclusion of neutral surface and/or rotary inertia. It is noticed that the inclusion of rotary inertia and neutral surface influences the eigenvalues of P-FG plate, and that cannot be discounted. The study also examines the influence of plate geometry, material gradient index, edge conditions, and elastic foundation modulus on the natural frequency of P-FG plate.

Broadband dynamic elastic moduli of honeycomb lattice materials: A generalized analytical approach

A generic analytical framework is proposed to obtain the dynamic elastic moduli of lattice materials under steady-state vibration conditions. The dynamic deformation behaviour of the individual beam elements of a lattice is distinct from the behaviour under a static condition. This leads to a completely different global deformation pattern of the lattice material and subsequently opens up a tremendous opportunity to modulate amplitude and phase of the elastic properties of lattices as a function of the ambient vibration. The dynamic stiffness approach proposed in this article precisely captures the sub-wavelength scale dynamics of the periodic network of beams in a lattice material using a single beam-like member. Here the dynamic stiffness matrix of a damped beam element based on the Timoshenko beam theory along with axial stretching is coupled with the unit cell-based approach to derive the most general closed-form analytical formulae for the elastic moduli of lattice materials across the whole frequency range. It is systematically shown how the general expressions of dynamic elastic moduli can be reduced to different special cases by neglecting axial and shear deformations under dynamic as well as classical static conditions. The significance of developing the dynamic stiffness approach compared to conventional dynamic finite element approach is highlighted by presenting detailed analytical derivations and representative numerical results. Further, it is shown how the analytical framework can be readily extended to lattices with non-prismatic beam elements with any spatial variation in geometry and intrinsic material properties. In general, research activities in the field of lattice metamaterials dealing with elastic properties revolve around intuitively designing the microstructural geometry of the lattice structure. Here we propose to couple the physics of deformation as a function of vibrating frequency along with the conventional approach of designing microstructural geometry to expand the effective design space significantly. The stretching-enriched physics of deformation in the lattice materials in addition to the bending and shear deformations under dynamic conditions lead to complex-valued elastic moduli due to the presence of damping in the constituent material. The amplitude, as well as the phase of effective elastic properties of lattice materials, can be quantified using the proposed approach. The dependence of Poisson's ratio on the intrinsic material physics in case of a geometrically regular lattice is found to be in contrary to the common notion that Poisson's ratios of perfectly periodic lattices are only the function of microstructural geometry. The generic analytical approach for analysing the elastic moduli is applicable to any form of two- or three-dimensional lattices, and any profile of the constituent beam-like elements (different cross-sections as well as spatially varying geometry and intrinsic material properties) through a wide range of frequency band. The closed-form expressions of elastic moduli derived in this article can be viewed as the broadband dynamic generalisation of the well-established classical expressions of elastic moduli under static loading, essentially adding a new exploitable dimension in the metamaterials research in terms of dynamics of the intrinsic material.

Direct Method for Generating Floor Response Spectra considering Soil–Structure Interaction

ABSTRACT A methodology is developed for generating floor response spectra (FRS) considering the effect of dynamic soil–structure interaction (SSI) based on the substructure technique and a recently-developed direct spectra-to-spectra method. Dynamic stiffness matrix of a three-dimensional structure with a rigid foundation subject to three translational and three rotational base excitations is derived, and is expressed in terms of the modal information of the structure. A transfer matrix, depending on the dynamic stiffness matrices of the structure-foundation system and generalized soil springs, is developed to modify the tri-directional response spectra at the foundation level of free-field (FIRS) in soil medium. The modified response spectra, called foundation level input response spectra (FLIRS), are used as the input to the fixed-base structure to generate FRS using the direct spectra-to-spectra method. Numerical examples of a reactor building and a service building in nuclear power plants demonstrate the accuracy and efficiency of the proposed method. FRS with % and % Non-Exceedance Probability (NEP) given by the proposed method agree well with the FRS obtained from a large number of time history analyses. Furthermore, the effect of SSI on FRS is studied, the physical meaning, and the advantages of the proposed method are highlighted.

An accurate frequency-domain model of water interaction with cylinders of arbitrary shape during earthquakes

An accurate frequency domain model is proposed to analyze the seismic response of uniform vertical cylinders with arbitrary cross section surrounded by water. According to the boundary conditions and using the variables separation method, the vertical modes of the hydrodynamic pressure are firstly obtained. Secondly, the three-dimensional wave equation can be simplified to a two-dimensional Helmholtz equation. Introducing the scaled boundary coordinate, a scaled boundary finite element (SBFE) equation which is a linear non-homogeneous second-order ordinary equation is derived by weighted residual method. The dynamic-stiffness matrix equation for the problem is furtherly derived. The continued fraction is acted as the solution of the dynamic-stiffness matrix for cylinder dynamic interaction of cylinder with infinite water. The coefficient matrices of the continued fraction are derived recursively from the SBFE equation of dynamic-stiffness. The accuracy of the present method is verified by comparing the hydrodynamic force on circular, elliptical and rectangle cylinders with the analytical or numerical solutions. Finally, the proposed model is used to analyze the natural frequency and seismic response of cylinders.

Dynamic Stiffness Formulation for Free Vibration of Truncated Conical Shell and Its Combinations with Uniform Boundary Restraints

This paper presents a dynamic stiffness formulation for the free vibration analysis of truncated conical shell and its combinations with uniform boundary restraints. The displacement fields are expressed as power series, and the coefficients of the series are obtained as recursion formula by substituting the power series into the governing equations. Then, the general solutions can be replaced by an algebraic sum which contains eight base functions, which can diminish the number of degrees of freedom directly. The dynamic stiffness matrix is formulated based on the relationship between the force and displacement along the boundary lines. In the formulation, arbitrary elastic boundary restraints can be realized by introducing four sets of boundary springs along the displacement directions at the boundary lines. The modeling methodology can be easily extended to the combinations of conical shells with different thickness and semivertex angles. The convergence and accuracy of the present formulation are demonstrated by comparing with the finite element method using several numerical examples. Effects of the elastic boundary condition and geometric dimension on the free vibration characteristics are investigated, and several representative mode shapes are depicted for illustrative purposes.

Free vibration analysis of an orthotropic plate by dynamic stiffness method and Wittrick-Williams algorithm

The Wittrick–Williams algorithm is used as the solution technique in the dynamic stiffness matrix to compute the natural frequencies of the orthotropic plate for different boundary conditions, aspect ratios, thickness ratios and modulus ratios. The corresponding dynamic stiffness matrix of the orthotropic plate is developed by classical plate theory and Hamilton’s principle. Levy type boundary conditions are considered in this paper. Results obtained are compared with published results and that obtained by finite element method using commercial software Ansys.

Investigating the effect of edge crack on the modal properties of composite wing using dynamic stiffness matrix

In this study free vibration analysis of a cracked Goland composite wing is investigated. The wing is modelled as a cantilevered beam based on Euler- Bernoulli equations. Also, composite material is modelled based on lamina fiber-reinforced. Edge crack is modelled by additional boundary conditions and local flexibility matrix in crack location, Castigliano's theorem and energy release rate formulation. Governing differential equations are extracted by Hamilton's principle. Using the separation of variables method, general solution in the normalized form for bending and torsion deflection is achieved then expressions for the cross-sectional rotation, the bending moment, the shear force and the torsional moment for the cantilevered beam are obtained. The cracked beam is modelled by separation of beam into two interconnected intact beams. Free vibration analysis of the beam is performed by applying boundary conditions at the fixed end, the free end, continuity conditions in the crack location of the beam and dynamic stiffness matrix determinant. Also, the effects of various parameters such as length and location of crack and fiber angle on natural frequencies and mode shapes are studied. Modal analysis results illustrate that natural frequencies and mode shapes are affected by depth and location of edge crack and coupling parameter.

Band-gap characteristics of elastic metamaterial plate with axial rod core by the finite element and spectral element hybrid method

A new kind of elastic metamaterial plate is designed to control the vibration and elastic wave propagation based on the band-gap mechanism. For the proposed elastic metamaterial plate, local resonant rods are periodically distributed between the two cover plates of the metamaterial plate structure. The dynamic stiffness matrix and the frequency response function of the structure under the time-harmonic excitation are obtained by the finite element and spectral element hybrid method (FE-SEHM). The experiment and simulation validations have been carried out. The effects of the material and structural parameters on the band-gap characteristics are analyzed. It is shown that the band-gaps can be computed more accurately and efficiently based on the FE-SEHM to avoid confusion between the pass-bands and band-gaps. Three band-gaps of the elastic metamaterial plate appear near the odd-order natural frequencies of the local resonant rods. Increasing the damping and mass of the local resonant rods results in the enlargement of the band-gap width, this enhances the capacity of the vibration reduction and sound isolation more effectively.