Usik Lee

Spectral element method in structural dynamics

Preface. Part One Introduction to the Spectral Element Method and Spectral Analysis of Signals. 1 Introduction. 1.1 Theoretical Background. 1.2 Historical Background. 2 Spectral Analysis of Signals. 2.1 Fourier Series. 2.2 Discrete Fourier Transform and the FFT. 2.3 Aliasing. 2.4 Leakage. 2.5 Picket-Fence Effect. 2.6 Zero Padding. 2.7 Gibbs Phenomenon. 2.8 General Procedure of DFT Processing. 2.9 DFTs of Typical Functions. Part Two Theory of Spectral Element Method. 3 Methods of Spectral Element Formulation. 3.1 Force-Displacement Relation Method. 3.2 Variational Method. 3.3 State-Vector Equation Method. 3.4 Reduction from the Finite Models. 4 Spectral Element Analysis Method. 4.1 Formulation of Spectral Element Equation. 4.2 Assembly and the Imposition of Boundary Conditions. 4.3 Eigenvalue Problem and Eigensolutions. 4.4 Dynamic Responses with Null Initial Conditions. 4.5 Dynamic Responses with Arbitrary Initial Conditions. 4.6 Dynamic Responses of Nonlinear Systems. Part Three Applications of Spectral Element Method. 5 Dynamics of Beams and Plates. 5.1 Beams. 5.2 Levy-Type Plates. 6 Flow-Induced Vibrations of Pipelines. 6.1 Theory of Pipe Dynamics. 6.2 Pipelines Conveying Internal Steady Fluid. 6.3 Pipelines Conveying Internal Unsteady Fluid. Appendix 6.A: Finite Element Matrices: Steady Fluid. Appendix 6.B: Finite Element Matrices: Unsteady Fluid. 7 Dynamics of Axially Moving Structures. 7.1 Axially Moving String. 7.2 Axially Moving Bernoulli-Euler Beam. 7.3 Axially Moving Timoshenko Beam. 7.4 Axially Moving Thin Plates. Appendix 7.A: Finite Element Matrices for Axially Moving String. Appendix 7.B: Finite Element Matrices for Axially Moving Bernoulli-Euler Beam. Appendix 7.C: Finite Element Matrices for Axially Moving Timoshenko Beam. Appendix 7.D: Finite Element Matrices for Axially Moving Plate. 8 Dynamics of Rotor Systems. 8.1 Governing Equations. 8.2 Spectral Element Modeling. 8.3 Finite Element Model. 8.4 Numerical Examples. Appendix 8.A: Finite Element Matrices for the Transverse Bending Vibration. 9 Dynamics of Multi-Layered Structures. 9.1 Elastic-Elastic Two-Layer Beams. 9.2 Elastic-Viscoelastic-elastic-Three-Layer (PCLD) Beams. Appendix 9.A: Finite Element Matrices for the Elastic-Elastic Two-Layer Beam. Appendix 9.B: Finite Element Matrices for the Elastic-VEM-Elastic Three-Layer Beam. 10 Dynamics of Smart Structures. 10.1 Elastic-Piezoelectric Two-Layer Beams. 10.2 Elastic-Viscoelastic-Piezoelctric Three-Layer (ACLD) Beams. 11 Dynamics of Composite Laminated Structures. 11.1 Theory of Composite Mechanics. 11.2 Equations of Motion for Composite Laminated Beams. 11.3 Dynamics of Axial-Bending-Shear Coupled Composite Beams. 11.4 Dynamics of Bending-Torsion-Shear Coupled Composite Beams. Appendix 11.A: Finite Element Matrices for Axial-Bending-Shear Coupled Composite Beams. Appendix 11.B: Finite Element Matrices for Bending-Torsion-Shear Coupled Composite Beams. 12 Dynamics of Periodic Lattice Structures. 12.1 Continuum Modeling Method. 12.2 Spectral Transfer Matrix Method. 13 Biomechanics: Blood Flow Analysis. 13.1 Governing Equations. 13.2 Spectral Element Modeling: I. Finite Element. 13.3 Spectral Element Modeling: II. Semi-Infinite Element. 13.4 Assembly of Spectral Elements. 13.5 Finite Element Model. 13.6 Numerical Examples. Appendix 13.A: Finite Element Model for the 1-D Blood Flow. 14 Identification of Structural Boundaries and Joints. 14.1 Identification of Non-Ideal Boundary Conditions. 14.2 Identification of Joints. 15 Identification of Structural Damage. 15.1 Spectral Element Modeling of a Damaged Structure. 15.2 Theory of Damage Identification. 15.3 Domain-Reduction Method. 16 Other Applications. 16.1 SEM-FEM Hybrid Method. 16.2 Identification of Impact Forces. 16.3 Other Applications. References. Index.

A frequency response function-based structural damage identification method

This paper introduces an frequency response function (FRF)-based structural damage identification method (SDIM) for beam structures. The damages within a beam structure are characterized by introducing a damage distribution function. It is shown that damages may induce the coupling between vibration modes. The effects of the damage-induced coupling of vibration modes and the higher vibration modes omitted in the analysis on the accuracy of the predicted vibration characteristics of damaged beams are numerically investigated. In the present SDIM, two feasible strategies are introduced to setup a well-posed damage identification problem. The first strategy is to obtain as many equations as possible from measured FRFs by varying excitation frequency as well as response measurement point. The second strategy is to reduce the domain of problem, which can be realized by the use of reduced-domain method introduced in this study. The feasibility of the present SDIM is verified through some numerically simulated damage identification tests.

A finite element for beams having segmented active constrained layers with frequency-dependent viscoelastics

A finite element for planar beams with active constrained layer (ACL) damping treatments is presented. Features of this non-shear locking element include a time-domain viscoelastic material model, and the ability to readily accommodate segmented (i.e. non-continuous) constraining layers. These features are potentially important in active control applications: the frequency-dependent stiffness and damping of the viscoelastic material directly affects system modal frequencies and damping; the high local damping of the viscoelastic layer can result in complex vibration modes and differences in the relative phase of vibration between points; and segmentation, an effective means of increasing passive damping in long- wavelength vibration modes, affords multiple control inputs and improved performance in an active constrained layer application. The anelastic displacement fields (ADF) method is used to implement the viscoelastic material model, enabling the straightforward development of time-domain finite elements. The performance of the finite element is verified through several sample modal analyses, including proportional-derivative control based on discrete strain sensing. Because of phasing associated with mode shapes, control using a single continuous ACL can be destabilizing. A segmented ACL is more robust than the continuous treatment, in that the damping of modes at least up to the number of independent patches is increased by control action.

The spectral element method in structural dynamics

The authors present a brief review of the spectral element method (SEM) in structural dynamics. A background discussion is included that provides a reference to previous works. The dynamic shape function approach of spectral element formulation is reviewed, and the state-vector approach is introduced for one-dimensional structures. The SEM for the forced vibrations of one-dimensional structures is briefly introduced in a general form for both the point and the distributed dynamic loads and is applied to Levy-type plates and smart beam structures. The accuracy of the spectral element is verified by comparing the SEM solutions with the solutions obtained by other methods including the conventional finite element method.

Spectral element modeling for the beams treated with active constrained layer damping

Abstract This paper introduces a spectrally formulated finite element for the beams with active constrained layer damping (ACLD). The spectral ACLD beam element is formulated from exact wave solutions of a set of fully coupled dynamic equations of motion. The fully coupled dynamic equations of motion are derived by using Hamilton’s principle, and they include the axial motion and rotary inertia of the viscoelastic layer. The dynamic responses obtained by the experiment, spectral element method and the finite element method are compared to evaluate the validity and accuracy of the present spectral ACLD beam element model. The control performances of an actively controlled ACLD beam predicted by using the present spectral element model and the conventional finite element model are also compared. The spectral ACLD beam element model is found to provide very reliable results when compared with the conventional finite element model.

Spectral analysis for the transverse vibration of an axially moving Timoshenko beam

The use of frequency-dependent spectral element matrix (or exact dynamic stiffness matrix) in structural dynamics is known to provide very accurate solutions, while reducing the number of degrees of freedom to resolve the computational and cost problems. Thus, in the present paper, the spectral element model is formulated for the axially moving Timoshenko beam under a uniform axial tension. The high accuracy of the present spectral element is then verified by comparing its solutions with the conventional finite element solutions and exact analytical solutions. The effects of the axial velocity and axial tension on the vibration characteristics, the dispersion relation, and the stability of a moving Timoshenko beam are analytically and numerically investigated.

Modal spectral element formulation for axially moving plates subjected to in-plane axial tension

The use of frequency-dependent spectral element matrix (or dynamic stiffness matrix) in structural dynamics is known to provide very accurate solutions, while reducing the number of degrees-of-freedom to resolve the computational and cost problems. Thus, in the present paper, the modal spectral element is formulated for thin plates moving with constant speed under a uniform in-plane axial tension. The concept of the Kantorovich method is used to formulate the modal spectral element matrix in the frequency-domain. The present modal spectral element is then evaluated by comparing its solutions with exact analytical solutions as well as with FEM solutions. The effects of the moving speed and the in-plane tension on the dynamic characteristics of a moving plate are investigated numerically.

Vibration analysis of one-dimensional structures using the spectral transfer matrix method

The spectral element matrix, often named the dynamic stiffness matrix, is known to provide the accurate dynamic characteristics of a structure because it is formed by exact shape functions. However, it is not always easy to derive the exact shape functions for any structure. Thus this paper first introduces a general approach to spectral element formulation for one-dimensional structures, in which the spectral element matrix is computed numerically directly from the transfer (or transition) matrix formulated from the state vector equation of motion of a structure. Next, by combining the promising features of the spectral element method (i.e., high accuracy) and the well-known transfer matrix method (i.e., high analysis efficiency for one-dimensional structures), a new solution approach named the spectral transfer matrix method (STMM) is introduced herein. Lastly a beam with periodic supports and a plane lattice structure with several beam-like periodic lattice substructures are considered as illustrative examples.

Revisiting the moving mass problem : Onset of separation between the mass and beam

In the moving mass problem, the interaction force between a moving mass and structure obviously depends on the velocity of moving mass and the flexibility of structure. Thus, in some situations, the interaction force may become zero to change its sign, which implies the onset of the separation between the moving mass and structure. Most investigations on this subject have missed or ignored the possibility of the onset of separation in solving the dynamic responses of structures excited by moving masses. Hence, this paper investigates the onset of the separation between the moving mass and beam, and then takes into account its effect in calculating the interaction forces and also in calculating the dynamic responses of the beams considered herein. It is shown that the separation between the moving mass and structure can occur more easily and has unnegligible effects on the dynamic responses of the structure as the mass ratio (M/ml) increases, especially at high velocity of moving mass. Thus, for accurate prediction of the dynamic response of structure excited by a moving mass, the effect of separation must be taken into account in the analysis.

Dynamics of elastic-piezoelectric two-layer beams using spectral element method

Abstract It is important to predict the dynamic characteristics of a piezoelectrically actuated beam very accurately for successful vibration controls. It has been well recognized that spectral elements provide very accurate solutions for such simple structures as beams. Thus, this paper introduces a spectral element method (SEM) and a spectral-element based modal analysis method (MAM) for elastic-piezoelectric two-layer beams. The axial-bending coupled equations of motion are derived first by using Hamilton’s principle and the spectral element matrix is formulated from the spectrally formulated exact eigenfunctions of the coupled governing equations. For MAM, the orthogonality of the eigenfunctions (i.e. natural modes) is proved. Present solution approaches are verified by comparing their results with the conventional FEM results. It is shown that the results by MAM and FEM converge to those by SEM as the number of superposed natural modes and the number of discretized finite elements are increased, respectively. It is also shown that, as the thickness of piezoelectric layer vanishes, the axial-bending coupled problems are decoupled to yield the solutions for two independent problems: the pure axial-motion problem and the pure bending-motion problem.