Dynamic Stiffness Matrix Method Research Articles

Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams

In this paper, the motion differential equations of the bi-directional functionally graded Timoshenko beam are established using Hamilton’s principle. The material properties of the beam change exponentially in both axial and thickness directions. Using the variable substitution method, state space differential equations of the structure are established. First, the dynamic stiffness matrix is formed using the traditional method. Then, the author proposes a new method to directly form an exact dynamic stiffness matrix by using state space differential equations, and this method is compared with the traditional dynamic stiffness matrix method. At the same time, the natural frequency of the structure is computed by combining the Wittrick–William algorithm with a non-iterative algorithm. The influence of gradient parameters α,β on the fundamental frequency, mode shape and frequency response function is analysed through the establishment of the dynamic stiffness matrix of the overall structure. Finally, using the Lagrange equation and the method of modal superposition, structural dynamic differential equations under a harmonic moving load are derived. Using the precise integration method, the dynamic response of the displacement is computed and the influence of gradient parameters α,β on the dynamic response is analysed.

Dynamic stiffness matrix of partial-interaction composite beams

Composite beams have a wide application in building and bridge engineering because of their advantages of mechanical properties, constructability and economic performance. Unlike static characteristics, the methods of studying the dynamic characteristics of partial-interaction composite beams were limited, especially dynamic stiffness matrix method. In this article, the dynamic stiffness matrix of partial-interaction composite beams was derived based on the assumption of the Euler–Bernoulli beam theory, and then it was used to predict the frequencies of the free vibration of the single-span composite beams with various boundary conditions or different axial forces. The corresponding vibration modes and buckling loads were also obtained. From the comparison with the existing results, the numerical results obtained by the proposed method agreed reasonably with those in the literatures. The dynamic stiffness matrix method is an accurate method which can determine natural vibration frequencies and vibration mode shapes in any precision theoretically. As a result, when the higher precision or natural frequencies of higher order are required, the dynamic stiffness matrix method is superior when compared to other approximate and numerical methods. The dynamic stiffness matrix method can also be combined with the finite-element method to calculate the free vibration frequencies and natural mode shapes of composite beams in complex conditions.

Element for Beam Dynamic Analysis Based on Analytical Deflection Trial Function

For beam dynamic finite element analysis, according to differential equation of motion of beam with distributed mass, general analytical solution of displacement equation for the beam vibration is obtained. By applying displacement element construction principle, the general solution of displacement equation is conversed to the mode expressed by beam end displacements. And taking the mode as displacement trial function, element stiffness matrix and element mass matrix for beam flexural vibration and axial vibration are established, respectively, by applying principle of minimum potential energy. After accurate integral, explicit form of element matrix is obtained. The comparison results show that the series of relative error between the solution of analytical trial function element and theoretical solution is about1×10-9and the accuracy and efficiency are superior to that of interpolation trial function element. The reason is that interpolation trial function cannot accurately simulate the displacement mode of vibrating beam. The accuracy of dynamic stiffness matrix method is almost identical with that of analytical trial function. But the application of dynamic stiffness matrix method in engineering is limited. The beam dynamic element obtained in this paper is analytical and accurate and can be applied in practice.

The dynamic, six-axis stiffness matrix testing of porcine spinal specimens

Background contextComplex testing protocols are required to fully understand the biomechanics of the spine. There remains limited data concerning the mechanical properties of spinal specimens under dynamic loading conditions in six axes. PurposeTo provide new data on the mechanical properties of functional spinal unit (FSU) and isolated disc (ISD) spinal specimens in 6 df. Study designDynamic, six-axis stiffness matrix testing of porcine lumbar spinal specimens. MethodsThe stiffness matrix testing of lumbar porcine FSU (n=6) and ISD (n=6) specimens was completed in a custom six-axis spine simulator using triangle wave cycles at a frequency of 0.1 Hz. Specimens were first tested without an axial preload, then with an axial preload of 500 N, with equilibration times of both 30 and 60 minutes. ResultsThe stiffness matrices were not symmetrical about the principal stiffness terms. The facets increased all the principal stiffness terms with the exception of axial compression-extension. Significant differences were detected in 15 stiffness terms because of the application of an axial preload in the ISD specimens, including an increase in all principal stiffness terms. There were limited differences in stiffness because of equilibration time of 30 and 60 minutes. ConclusionsThe assumption of stiffness matrix symmetry used in many previous studies is not valid. The biomechanical testing of spinal specimens should be completed in 6 df, at physiologic loading rates, and incorporate the application of an axial preload. The present study has provided new data on the mechanical properties of spinal specimens and demonstrates that the dynamic stiffness matrix method provides a means to more fully understand the natural spine and quantitatively assess spinal instrumentation.

Wave interpretation of numerical results for the vibration in thin conical shells

The dynamic behaviour of thin conical shells can be analysed using a number of numerical methods. Although the overall vibration response of shells has been thoroughly studied using such methods, their physical insight is limited. The purpose of this paper is to interpret some of these numerical results in terms of waves, using the wave finite element, WFE, method. The forced response of a thin conical shell at different frequencies is first calculated using the dynamic stiffness matrix method. Then, a wave finite element analysis is used to calculate the wave properties of the shell, in terms of wave type and wavenumber, as a function of position along it. By decomposing the overall results from the dynamic stiffness matrix analysis, the responses of the shell can then be interpreted in terms of wave propagation. A simplified theoretical analysis of the waves in the thin conical shell is also presented in terms of the spatially-varying ring frequency, which provides a straightforward interpretation of the wave approach. The WFE method provides a way to study the types of wave that travel in thin conical shell structures and to decompose the response of the numerical models into the components due to each of these waves. In this way the insight provided by the wave approach allows us to analyse the significance of different waves in the overall response and study how they interact, in particular illustrating the conversion of one wave type into another along the length of the conical shell.

On the Free Vibration Modeling of Spindle Systems: A Calibrated Dynamic Stiffness Matrix

The effect of bearings on the vibrational behavior of machine tool spindles is investigated. This is done through the development of a calibrated dynamic stiffness matrix (CDSM) method, where the bearings flexibility is represented by massless linear spring elements with tuneable stiffness. A dedicated MATLAB code is written to develop and to assemble the element stiffness matrices for the system’s multiple components and to apply the boundary conditions. The developed method is applied to an illustrative example of spindle system. When the spindle bearings are modeled as simply supported boundary conditions, the DSM model results in a fundamental frequency much higher than the system’s nominal value. The simply supported boundary conditions are then replaced by linear spring elements, and the spring constants are adjusted such that the resulting calibrated CDSM model leads to the nominal fundamental frequency of the spindle system. The spindle frequency results are also validated against the experimental data. The proposed method can be effectively applied to predict the vibration characteristics of spindle systems supported by bearings.

Experimental evaluation and theoretical analysis of multi-layered road cumulative deformation under dynamic loads

Excessive deformation of multi-layered road structure induced by vehicle traffic loading is one of the main reasons leading to pavement failure. In this paper, the elastic response and cumulative deformation of a multi-layered road structure under dynamic loads are investigated through experimental tests and theoretical analysis. First, a physical model with thin asphalt surfacing layer, cement stabilised gravel road base and layered decomposed granite soil subgrade is built to better understand the dynamic behaviour of a multi-layered road structure system. Experiments are performed to determine the elastic responses (vertical stress and displacement) and cumulative plastic deformation at different depths of road structures. A calculation method of the dynamic stiffness matrix is developed on the basis of an assembly process compatible with good numerical efficiency, and results from the calculations are verified using model experiment data. Then, the dynamic deviator stress is derived according to the dynamic stiffness matrix method and is used to formulate a cumulative deformation computational model considering both road infrastructure and its soft subsoil. Moreover, the cumulative deformation of a road on soft subsoil under repeated traffic load is analysed for different material properties of the road layers using the presented method.

Research on Large Bulk Carrier Hull Production Design Process Planning Based on Dynamic Stiffness Matrix Method

According to the shortcomings such as long cycle, large consumption in view of the traditional method of shipbuilding, introduce the method named collaborative design, build hull production design process model. First, decompose hull production design process, then introduce the method of DSM-matrix to describe hull production design task, build the hull production design process model, then based on the model, develop collaborative design system. And use the system to make the hull production design process more reasonable. At last successfully apply them to the collaborative design system. The results suggest that the hull production design method based on DSM-matrix can effectively shorten the manufacturing cycle and save material.

Vibration Modeling of Machine Tool Spindles: A Calibrated Dynamic Stiffness Matrix Method

The effects of spindles vibrational behavior on the stability lobes and the Chatter behavior of machine tools have been established. The service life has been observed to reducethe system natural frequencies. An analytical model of a multi-segment spinning spindle, based on the Dynamic Stiffness Matrix (DSM) formulation, exact within the limits of the Euler-Bernoulli beam bending theory, is developed. The system exhibits coupled Bending-Bending (B-B) vibration and its natural frequencies are found to decrease with increasing spinning speed. The bearings were included in the model usingboth rigid, simply supported, frictionless pins and flexible linear spring elements. The linear spring element stiffness is then calibrated so that the fundamental frequency of the system matches the nominal value.

Dynamic stiffness matrix method for axially moving micro-beam

In this paper the dynamic stiffness matrix method was used for the free vibration analysis of axially moving micro beam with constant velocity. The extended Hamilton's principle was employed to derive the governing differential equation of the problem using the modified couple stress theory. The dynamic stiffness matrix of the moving micro beam was evaluated using appropriate expressions of the shear force and bending moment according to the Euler-Bernoulli beam theory. The effects of the beam size and axial velocity on the dynamic characteristic of the moving beam were investigated. The natural frequencies and critical velocity of the axially moving micro beam were also computed for two different end conditions.

Homogeneous trees of second order Sturm–Liouville equations: A general theory and program

Quantum graph problems occur in many disciplines of science and engineering and they can be solved by viewing the problem as a structural engineering one. The Sturm–Liouville operator acting on a tree is an example of a quantum graph and the structural engineering analogy is the axial vibration of an assembly of bars connected together with a tree topology. Using the dynamic stiffness matrix method the natural frequencies of the system can be determined which are analogous to the eigenvalues of the quantum graph. Theory is presented that yields exact solutions to the Sturm–Liouville problem on homogeneous trees. This is accompanied by an extremely efficient and compact computer program that implements the theory. An understanding of the former is enhanced by recourse to a structural mechanics analogy, while the latter program is fully annotated and explained for those who might wish to extend its capability. In addition, the use of the program as a ‘black box’ is fully described and a small parametric study is undertaken to confirm the accuracy of the approach and indicate its range of application including the computation of negative eigenvalues.

Free vibrations of Timoshenko beam-columns on Pasternak foundations

In this paper the free vibration and the stability of axially loaded Timoshenko beams on elastic foundation are analyzed through the dynamic stiffness matrix method. The study is carried out considering a two–parameter elastic soil which contains the characteristics of both the Winkler and the Pasternak foundations. The equation of motion and the exact dynamic stiffness matrix of a beam–column including the effect of shear deformation and rotatory inertia is derived. The eigenproblem of the beam–column is therefore solved through the application of the general Wittrick and Williams algorithm. The effects of soil characteristics as well as shear deformations and rotatory inertia on the natural frequencies of the beam-column are investigated. An extensive parametric study highlights the main parameters that characterize the dynamical behavior of the considered beam–column.

Dispersion relations and modes of wave propagation in inclusion-reinforced composite plates

This paper is intended to examine the effect of inclusion shapes, inclusion contents, inclusion elastic constants, and plate thickness on the dispersion relations and modes of wave propagation in inclusion-reinforced composite plates. The shape of inclusion is modeled as spheroid that enables the composite reinforcement geometrical configurations ranging from sphere to short and continuous fiber. Mori–Tanaka mean-field theory is used to predict the effective elastic moduli of the composite plate explicitly. The effective elastic moduli are able to elucidate the effect of inclusion’s shape, stiffness, and volume fraction on the composite’s anisotropic elastic behavior. The resulting moduli are then used to determine the dispersion relations and the modal patterns of Lamb waves using the dynamic stiffness matrix method. The types (symmetric or antisymmetric) of Lamb waves in an isotropic plate can be classified according to the wave motions are symmetrical or antisymmetric about the midplane of the plate. Classifying the wave type in an anisotropic plate is not as simple as that in an isotropic plate, and has not received proper attention in the literature. The wave types and orders are identified by analyzing the dispersion curves and inspecting the calculated modal patterns, and the results indicate that the Lamb waves in an orthotropic composite plate can also be classified as either symmetric or antisymmetric waves. It is also found that the inclusion contents, aspect ratios and plate thickness affect propagation velocities, higher-order mode cutoff frequencies, and modal patterns. Propagation speed is generally increased with the aspect ratio, e.g., using longer fibers generally results in a higher propagation speed.

A Dynamic Stiffness Element for Free Vibration Analysis of Delaminated Layered Beams

A dynamic stiffness element for flexural vibration analysis of delaminated multilayer beams is developed and subsequently used to investigate the natural frequencies and modes of two-layer beam configurations. Using the Euler-Bernoulli bending beam theory, the governing differential equations are exploited and representative, frequency-dependent, field variables are chosen based on the closed form solution to these equations. The boundary conditions are then imposed to formulate the dynamic stiffness matrix (DSM), which relates harmonically varying loads to harmonically varying displacements at the beam ends. The bending vibration of an illustrative example problem, characterized by delamination zone of variable length, is investigated. Two computer codes, based on the conventional Finite Element Method (FEM) and the analytical solutions reported in the literature, are also developed and used for comparison. The intact and defective beam natural frequencies and modes obtained from the proposed DSM method are presented along with the FEM and analytical results and those available in the literature.

Dynamic Finite Element Analysis of Extensional-Torsional Coupled Vibration in Nonuniform Composite Beams

The application of a Dynamic Finite Element (DFE) technique to the extensional-torsional free vibration analysis of nonuniform composite beams, in the absence of flexural coupling, is presented. The proposed method is a fusion of the Galerkin weighted residual formulation and the Dynamic Stiffness Matrix (DSM) method, where the basis functions of approximation space are assumed to be the closed form solutions of the differential equations governing uncoupled extensional and torsional vibrations of the beam. The use of resulting dynamic trigonometric interpolation (shape) functions leads to a frequency dependent stiffness matrix, representing both mass and stiffness properties of the beam element. Assembly of the element matrices and the application of the boundary conditions then leads to a frequency dependent nonlinear eigenproblem, which is solved to evaluate the system natural frequencies and modes. Two illustrative examples of uniform and tapered cantilevered, Circumferentially Uniform Stiffness (CUS), hollow, composite beams are presented. The influence of ply fibre-angle on the natural frequencies is also studied. The correctness of the theory and the superiority of the proposed DFE over the contrasting DSM and conventional FEM methods are confirmed by the published results and numerical checks. The discussion of results is followed by some concluding remarks.

Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory

The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.

Modeling of Coriolis mass flow meter of a general plane-shape pipe

The vibration of a general plane tube with a flowing fluid, which is the measuring element in a Coriolis mass flow meter (CMF), is studied. The dynamic stiffness matrix method is used to model such a tube. The tube is divided into straight and circular elements. The elements dynamic stiffness matrices are derived from the equations of motion. By assembling the elements matrices into a global matrix the natural frequencies are obtained. The mode shapes are obtained by applying the boundary conditions at the supports and the compatibility conditions at the nodes. The effects of the flow velocity on the natural frequency and the relative phase difference are modeled. The method is applied to different tube shapes. The results are compared to the published data which reveals good agreement.

Free Vibration Analysis of Sandwich Beams: A Dynamic Finite Element

The free vibration analysis of three-layered symmetric sandwich beams is studied. The governing differential equations of motion in free vibration are coupled both in axial and bending deformations. The weighted residual method is used and the resulting weak integral form of the governing equations is discretized using the Finite Element Method (FEM) formulation. The dynamic, frequency-dependent, trigonometric shape functions, derived from the solution of the uncoupled equations, are then used to develop the element matrices. The assembly of resulting Dynamic Finite Element (DFE) matrices leads to a nonlinear eigenvalue problem. A determinant search method is then used to compute the coupled natural frequencies and modes of a symmetric, thin face-layer sandwich beam. The DFE numerical results for the first four bending-axial coupled modes, obtained from a 20-element mesh, show good agreement with FEM, exact Dynamic Stiffness Matrix (DSM) method, and other results available in the literature. All the four modes are found to be dominated by bending deformation. The discussion of results is followed by some concluding remarks.

Dynamic analysis of gear pairs with a gross motion effect using the dynamic stiffness matrix method

A continuum approach using the dynamic stiffness matrix (DSM) method (DSMM) is presented for the dynamic analyses of spur gear pairs incorporating the gross motion effect. A continuum dynamic model of spur gears was formulated based on the dynamic stiffness using the non-uniform Timoshenko beam theory. A systematic DSM for a gear pair was assembled including the elemental DSMs of two mating gears and non-linear Hertzian contact stiffness. The DSMs were updated at each computing step to reflect the time-varying property due to the moving mesh points. A bisection procedure was used to calculate the natural frequencies as the systematic DSM of the gear pair became singular. The mass and stiffness matrices were calculated by taking a numerical derivative of the systematic DSM. Finally, the gear dynamic responses caused by the external torque and gross motion excitation resulted. The effect of gross motion is also discussed.

Vibration Analysis of Cracked Stepped Laminated Composite Beams

The free vibration of piecewise uniform defective laminated composite beams is investigated. The governing differential equations of motion are coupled both in torsional and bending deformations. A Dynamic Finite Cracked Element (DFCE) is developed and is applied to slender beams, characterized by an offset between inertial and bending axes. The hybrid DFCE is a combination of the conventional Finite Element Method (FEM) formulation and frequencydependent interpolation functions stemmed from the exact Dynamic Stiffness Matrix (DSM) method. The defect, a through-thickness edge crack, is then represented by a set of stiffness terms evaluated from the beam compliance matrix at the crack location. A number of stepped beam configurations are investigated by reducing the base, thickness, or both. The natural frequencies and modes of free vibration of the beams are examined for single through-thickness edge crack configurations.