Dynamic Stiffness Matrices Research Articles

Predicting vibration transmission across junctions using diffuse field reciprocity

Predicting the sound insulation of an engineering system is a complex problem since not only the direct path through a separating element but also the flanking transmission paths can largely influence the sound insulation of the system. When conventionally analyzing flanking transmission, a diffuse field is assumed in the walls and floors, which are modelled as plates. The junction connecting the walls and floors is assumed to be of infinite extent and the transmission of vibration across the junction is calculated by integrating over all possible angles of incidence. Due to the limitations of the conventional approach, a new approach based on diffuse field reciprocity is proposed. The diffuse field reciprocity relationship relates the vibration transmission to the direct field of a diffuse subsystem to the direct field dynamic stiffness of the subsystem, i.e., the dynamic stiffness of the equivalent infinite subsystem as observed at the junction. The direct field dynamic stiffness matrices of thin, isotropic, elastic plates can be analytically derived. For more complex walls or floors a possible approach is to calculate the direct field dynamic stiffness using finite elements and perfectly matched layers. The perfectly matched layer surrounding the finite element model absorbs the wave propagating outwards from the bounded domain, thus simulating an infinite subsystem.

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.

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.

Dynamic stiffness formulation for transverse and in-plane vibration of rectangular plates with arbitrary boundary conditions based on a generalized superposition method

Dynamic stiffness formulation is proposed in this paper for both transverse and in-plane vibration of rectangular plates that account for arbitrary boundary conditions. A generalized superposition method is developed to obtain the homogeneous solutions for the governing equations of both transverse and in-plane vibration. Consequently, the dynamic stiffness matrices are formed in a more straightforward way by projection method, the dimensions of which are greatly reduced in comparison with those from the conventional Gorman’s superposition method. The finite element technique is utilized to assemble local stiffness matrix into global coordinates so as to address the dynamics of plate assemblies. Various types of plate-like structures are investigated by the proposed method, through which excellent agreement is found between our results and those from finite element method. The effectiveness, accuracy and convergence of the proposed DSM for both transverse and in-plane vibration are proved in several numerical examples, which demonstrates the proposed DSM is an excellent alternative to the existing DSM.

Transient analysis of three‐dimensional dynamic interaction between multilayered soil and rigid foundation

AbstractThe study of dynamic soil‐structure interaction is significant to civil engineering applications, such as machine foundation vibration, traffic‐induced vibration, and seismic dynamic response. The scaled boundary finite element method (SBFEM) is a semi‐analytical algorithm, which is used to solve the dynamic response of a three‐dimensional infinite soil. It can automatically satisfy the radiation boundary condition at infinity. Based on the dynamic stiffness matrix equation obtained by the modified SBFEM, a continued fraction algorithm is proposed to solve the dynamic stiffness matrix of layered soil in the frequency‐domain. Then, the SBFEM was coupled with the finite element method (FEM) at the interface to solve the dynamic stiffness matrices of the rigid surface/buried foundation. Finally, the mixed‐variable algorithm was used to solve the three‐dimensional transient dynamic response of the foundation in the time domain. Numerical examples were performed to verify the accuracy of the proposed algorithm in solving the dynamic stiffness matrix of the infinite domain in the frequency domain and the dynamic transient displacement response of the foundation in the time domain. Compared with the previous numerical integration technique, the dynamic stiffness matrix in the frequency domain calculated by using the proposed algorithm has higher accuracy and higher efficiency.

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.

Simple discrete models for dynamic structure-soil-structure interaction analysis

Simple models are presented for dynamic analyses of Structure-Soil-Structure Interaction (SSSI). Structures built on rigid, circular surface foundations resting on a homogeneous, elastic soil half-space are considered. The proposed models consist of discrete mass and spring elements that represent inertial, stiffness, and damping characteristics of an SSSI system. Frequency-dependent springs are developed to simulate Foundation-Soil Interaction (FSI) and Cross-Interaction (CI) between adjacent foundations, both being affected by the number and arrangement of neighbouring foundations. The stated springs are developed using a novel approach that approximates the through-the-soil interaction between adjacent footings using existing solutions to soil-structure interaction (SSI) analyses. Static stiffness matrices are obtained by means of compliance matrices using well known analytical solutions of surface displacement field to a ‘single-disk’ problem and a suggested displacement averaging rule. Dynamic stiffness matrices are created by applying modification factors to their static counterparts. To this end, a practical expression for dynamic modification factors that relate static and dynamic amplitudes of surface displacement field is provided. Equations of motion are formulated and solved in both time and frequency domains, taking into account time-lagging effects on input motion at separate foundations due to their vibration and wave passage. Validation results against more advanced numerical models prove the effectiveness of the proposed models. Finally, the limitations and potential improvements on the models are discussed.

Dynamic stiffness approach to vibration transmission within a beam structure carrying spring–mass systems

The dynamic stiffness method is developed for the dynamics of a beam structure carrying multiple spring−mass systems. Based on classical Bernoulli–Euler beam theory, three types of vibrations, namely, bending, longitudinal and torsional motions, are formulated in terms of dynamic stiffness matrix. Similar to finite element technique, the local dynamic stiffness matrices for individual beams and spring−mass systems are assembled into global dynamic stiffness matrices so as to address vibration transmission from machines to flexible beamlike foundations. Using our proposed method, the vibration transmission within a beam frame, carrying multiple spring−mass systems is addressed. Through numerical analysis, the calculated vibration responses agree well with those from finite element method, which demonstrates that dynamic stiffness formulation has great potential in modeling the dynamics of built-up beam structures that supports machinery, especially in characterizing vibration isolation due to wave reflections, wave conversions, and other underlying mechanisms.

Dynamic Response of 3D Surface/Embedded Rigid Foundations of Arbitrary Shapes on Multi-Layered Soils in Time Domain

Significant differences between the predicted and measured dynamic response of 3D rigid foundations on multi-layered soils in the time domain were identified due to the existence of uncertainties, which makes the issue a complicated one. In this study, a numerical method was developed to determine the dynamic responses of 3D rigid surfaces and embedded foundations of arbitrary shapes that are bonded to a multi-layered soil in the time domain. First, the dynamic stiffness matrices of the rigid foundations in the frequency domain are calculated via integral domain transformation. Secondly, a dynamic stiffness equation for rigid foundations in the time domain is established via the mixed variables formulation, which is based on the discrete dynamic stiffness matrices in the frequency domain. The proposed method can be applied to the treatment of systems with multiple degrees of freedom without losing the true information that concerns the coupling characteristics. Numerical examples are presented to demonstrate the accuracy of the proposed method for predicting the horizontal, vertical, rocking, and torsional vibrations. Further, a parametric study was carried out to provide insight into the dynamic behavior of the soil–foundation interaction (SFI) while considering soil nonhomogeneity. The results indicate that the elastic modulus of the soil has a significant impact on the dynamic responses of the rigid foundation. Finally, a numerical example of a rigid foundation resting on a six-layered, semi-infinite soil demonstrates that the proposed method can be used to deal with multi-layered media in the time domain in a relatively easy way.

2D nonlinear inversion of bedrock motion from the surface motion of a layered half-space

This paper proposes a method for two-dimensional (2D) nonlinear inversion of bedrock earthquake motion from the surface motion of a layered half-space. The exact dynamic-stiffness matrices of soil layer and half-space are used for the inversion of bedrock motion and incident angle from both the surface horizontal motion and vertical motion, and the equivalent linearization approach is used to consider the nonlinearity of the soil layers. The method of inversion is verified by an example of obliquely incident El Centro Earthquake record at bedrock. It is shown that, the bedrock motion and incident angle can be accurately inversed from both the horizontal motion and vertical motion. The one-dimensional (1D) traditional inversion of bedrock horizontal motion from only surface horizontal motion in literatures may cause large error due to ignoring wave mode conversion, because there is vertical motion accompanying two surface horizontal motions in real earthquake records anyway. The error in 1D inversion depends on the ratio of surface vertical motion to horizontal motion, and the larger the surface vertical motion, the larger the error.

Site response analysis in cross-anisotropic alluvial deposits subjected to inclined incident SH wave

Elastic site response analysis in cross-anisotropic deposits has been elaborated due to incidence of inclined SH wave front. A site consisting of a single cross-anisotropic soil layer rested on a half-space representing the bedrock is examined, subjected to inclined SH wave and the out-of-plane displacement amplitude is obtained by assembling the dynamic stiffness matrices of the layer and the underlying bedrock. The effects of the cross-anisotropy substantiated in form of different horizontal and vertical shear moduli were investigated and discussed accordingly. Results showed that anisotropy has great effect on the site response; however, this effect becomes immaterial when the incidence angles are almost sub-horizontal.

Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions

Axially moving beams are often discussed with several classic boundary conditions, such as simply-supported ends, fixed ends, and free ends. Here, axially moving beams with generalized boundary conditions are discussed for the first time. The beam is supported by torsional springs and vertical springs at both ends. By modifying the stiffness of the springs, generalized boundaries can replace those classical boundaries. Dynamic stiffness matrices are, respectively, established for axially moving Timoshenko beams and Euler-Bernoulli (EB) beams with generalized boundaries. In order to verify the applicability of the EB model, the natural frequencies of the axially moving Timoshenko beam and EB beam are compared. Furthermore, the effects of constrained spring stiffness on the vibration frequencies of the axially moving beam are studied. Interestingly, it can be found that the critical speed of the axially moving beam does not change with the vertical spring stiffness. In addition, both the moving speed and elastic boundaries make the Timoshenko beam theory more needed. The validity of the dynamic stiffness method is demonstrated by using numerical simulation.

A numerical simulation model for the vortex induced vibration of flexible risers using dynamic stiffness matrices

In this study, a time-domain numerical method using dynamic stiffness matrices is proposed to obtain the characteristics of the vortex induced vibration (VIV) of a flexible riser. Due to the large ratio of length to diameter, a marine riser is approximated as a cable model. Based on the finite volume method (FVM), the cable model of the riseris dividedinto a certain number of segments. The dynamic stiffness matrix of each segment reflecting the time varying strain energy and bending stiffness are updated in calculation. The total dynamic stiffness matrix of the riser is then obtained by the superposition characteristics of the finite volume method. In the nonlinear governing equation of riser's vibration, the hydrodynamic forces are calculated by the instantaneous vorticity conserved boundary condition (IVCBC) method and the equation is solved by the step-by-step time integration method to obtain the accelerations, velocities and displacements of the riser. Validation of this numerical approach is carried out by comparing the static balance and the dynamic responses with quantities in classical experimental cases for a riser in uniform flow. The results show that:when the vibration of riser is high order mode vibration, the dynamic stiffness matrix cannot be ignored. The vortex induced vibration of the given flexible riser has a strong coupling effect between the deformation of riser and flow field. In the vibtation of the riser, there exists the multiple-frequency effect, numely, adominant frequency co-exists with some subordinate frequencies, hence the shedding frequencies can no longer be satisfied with the distributions of the Strauhal number. The wake effect onthe vortex shedding has a stronger 3-dimensional effect at the locations of smaller lateral vibrations. The numerical simulations also reveal that the vibration mode increases with the flow velocity.

Resonance risk and global sensitivity analysis of a straight–curved combination pipe based on active learning Kriging model

The resonance failure of straight–curved combination pipes conveying fluid which are widely used in engineering is becoming a serious issue. But there are only few studies available on the resonance failure of combination pipes. The resonance failure probability and global sensitivity analysis of straight–curved combination pipes conveying fluid are studied by the active learning Kriging method proposed in this article. Based on the Euler–Bernoulli beam theory, the dynamic stiffness matrices of straight and curved pipes are derived in the local coordinate system, respectively. Then the dynamic stiffness matrix and characteristic equation of a straight–curved combination pipe conveying fluid are assembled under a global coordinate system. The natural frequency is calculated based on the characteristic equation. A resonance failure performance function is established based on the resonance failure mechanism and relative criterions. The active learning Kriging model based on expected risk function is introduced for calculating the resonance failure probability and moment-independent global sensitivity analysis index. The importance rankings of input variables are obtained with different velocities. According to the results, it is shown that the method proposed in this article provides a lot of guidance for resonance reliability analysis and anti-resonance design in combination pipes conveying fluid.

Non-linear parametric vibration responses of composite plates on Pasternak foundations using extended dynamic stiffness method

ABSTRACT In this work, the non-linear parametric vibration responses of rectangular laminated composite plates subjected on Pasternak foundation, under in-plane periodic compressive load applied along two opposite edges, are theoretically studied using the extended dynamic stiffness method. The authors present the new method to establish the exact dynamic stiffness matrices of rectangular laminated composite plates subjected to in-plane periodic forces based on general Von-Kármán large deflection plate theory. A set of second-order ordinary differential non-linear equations of extended Mathieu–Hill type with periodic coefficients is formed to determine the regions of dynamic instability and non-linear parametric vibration responses based on Bolotin method. The effects of various parameters such as static load, dynamic load, aspect ratio, boundary conditions, orthotropy and foundation stiffness factor on the dynamic instability regions and non-linear parametric vibration characteristics are investigated and discussed. For linear theory, the present results are found to be in agreement with available conclusions in the literature which was obtained based on the linear analysis.

Finite element free and forced vibration analysis of gradient elastic beam structures

The dynamic stiffness matrix of a gradient elastic flexural Bernoulli–Euler beam finite element is analytically constructed with the aid of the basic and governing equations of motion in the frequency domain. The simple gradient theory of elasticity is used with just one material constant (internal length) in addition to the classical moduli. The flexural element has one node at every end with three degrees of freedom per node, i.e., the displacement, the slope, and the curvature. Use of this dynamic stiffness matrix for a plane system of beams enables one by a finite element analysis to determine its dynamic response harmonically varying with time external load or the natural frequencies and modal shapes of that system. The response to transient loading is obtained with the aid of Laplace transform with respect to time. A stiffness matrix is constructed in the transformed domain, the problem is formulated and solved by the finite element method, and the time domain response is finally obtained by a time domain inversion of the transform solution. Because the exact solution of the governing equation of motion in the frequency domain is used as the displacement function, the resulting dynamic stiffness matrices and the obtained structural response or natural frequencies and modal shapes are also exact. Examples are presented to illustrate the method and demonstrate its advantages. The effects of the microstructure on the dynamic behavior of beam structures are also determined.

A Mixed Analytical-Numerical Method for the Vibro-Acoustic Analysis of an Underwater Ring-Stiffened Cylindrical Shell Containing Substructures

A new sono-elastic substructure method is proposed in this paper to improve the computational efficiency of the hull-substructure coupled and fluid-structure interacted vibration and acoustic radiation of a submerged cylindrical-shell-type vehicle. The typical part of the vehicle structure is divided into the main hull and the internal substructures. The fluid-structure interaction problem of the main hull is solved by an analytical method based on the simplified model of a single-hull ring-stiffened cylindrical shell simply supported at both ends. Meanwhile, the substructures are numerically modeled through the Finite Element Method, with the condensed dynamic stiffness matrices of them obtained via the Superelement Method of Modal Synthesis. The main hull and the internal substructures are then integrated according to the boundary compatibility conditions at the connecting parts. Thus, a Mixed Analytical-Numerical Substructure (MANS) method is formulated. The applicability of this method is validated by two numerical examples as well as the test results of a large-scale submerged structural model. It is shown that the MANS method is suitable for the prediction of vibration and acoustic radiation of typical cylindrical-shell-type submerged structures in the medium frequency region.

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.

Wave and finite element method for predicting sound transmission through finite multi-layered structures with fluid layers

In this paper, a modelling strategy is proposed to predict sound transmission loss (STL) of multi-layered panels with fluid layers based on a wave and finite element method. This method starts with meshing a small segment of each solid layer through the thickness to find the mass and stiffness matrices using a conventional finite element method. The acoustic pressures applied on the structure by the fluids surrounding and inside the structure are discretized into external nodal forces. By post-processing the matrices using equilibrium conditions and periodic structure theory, the spectral dynamic stiffness matrices (DSM) of solid layers are derived in the wavenumber domain. By assembling these matrices with those of fluid layers, which are derived analytically, a total spectral DSM is found and used to calculate the response to incident acoustic waves. To take account of the effect of the finite size of a structure, a spatial windowing technique is applied. Various numerical examples are presented to validate this method. For simple structures, analytical methods are available for comparison. For complicated structures, using analytical methods is difficult and the proposed method becomes valuable. Results are compared with measurements. The method is seen to give accurate predictions at little computational cost.

Harmonic response analysis of coupled plate structures using the dynamic stiffness method

In this study, the dynamic stiffness method (DSM) is applied to study the free and forced vibration behaviors of thin, three-dimensionally coupled plate structures. Both the flexural and in-plane vibrations are taken into consideration. In the formulation, the coupled plate structures are divided into several sub-plates, then the dynamic stiffness matrices of the sub-plates are derived separately by combining the superposition method with the projection method. According to the geometrical coupled condition between the sub-plates, the dynamic stiffness matrix of the whole coupled structure is assembled. To validate the present method, three numerical examples of coupled plate structures subjected to external excitations are performed and analyzed. The accuracy and reliability of the present method are demonstrated by comparing the present results with those obtained by the FEM. Then, the effects of the directions of external harmonic excitations and the coupled angle between the coupled plates on the dynamic responses are investigated. Furthermore, a computational efficiency study is given to illustrate the convergence and accuracy of the DSM.