Joohong Kim

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.

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.

Dynamics of Branched Pipeline Systems Conveying Internal Unsteady Flow

The pipeline system conveying high pressurized unsteady internal flow may experience severe transient vibrations due to the fluid-pipe interaction under the time-varying conditions imposed by the pump and valve operations. In the present work, a set of fully coupled dynamic equations of motion for the pipeline system are developed to include the effect of the circumferential strain due to the internal fluid pressure. A finite element formulation for the fully coupled dynamic equations of motion is introduced and applied to several sample pipeline systems. The connectivity conditions for both fluid and structural variables at the junction of a branched pipeline system are properly incorporated in the finite element formulation. To ensure the validity and accuracy of the present theory of pipedynamics, the same pipeline system considered in a reference work is revisited and the present numerical results are compared with those given in the reference work. A series pipeline system with high reservoir head is then analyzed to investigate the effect of the additional linear/nonlinear coupling terms in the present pipedynamic theory. Numerical tests show that the nonlinear coupling terms may become significant at high fluid pressure and velocity.

Determination of Nonideal Beam Boundary Conditions: A Spectral Element Approach

A spectral element approach to determine the nonideal or unknown structural boundary conditions of beams is introduced. The nonideal structural boundary conditions are represented by the frequency-dependent effective boundary springs: transverse springs and rotational springs. The effective boundary spring constants are then determined from the measured frequency response functions in conjunction with the spectral element method. Experiments are conducted for the one-end-supported and the two-end-supported beams to verify the present approach of boundary conditions identification. The analytical predictions obtained by using the identified boundary conditions are very close to the experiment measurements.

Development of a Wittrick–Williams algorithm for the spectral element model of elastic–piezoelectric two-layer active beams

Abstract In this paper, a Wittrick–Williams algorithm is developed for the elastic–piezoelectric two-layer active beams. The exact dynamic stiffness matrix (or spectral element matrix) is used for the development. This algorithm may help calculate all the required natural frequencies, which lie below any chosen frequency, without the possibility of missing any due to close grouping or due to the sign change of the determinant of spectral element matrix via infinity instead of via zero. The uniform and partially patched active beams are considered as the illustrative examples to confirm the present algorithm.

Spectral Element Model for the 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 moving speed and axial tension on the vibration characteristics, the dispersion relation, and the stability of a moving Timoshenko beam are analytically and numerically investigated.

SPECTRAL ELEMENT MODELING OF THE BEAMS WITH ACTIVE CONSTRAINED LAYER DAMPING

This paper introduces a spectrally formulated finite element of the beams with active constrained layer damping (ACLD). The spectral element is formulated from the exact eigenfunctions of a set of tilly coupled equations of motion derived by using Hamilton’s principle. The coupled equations of motion include the axial motion and rotatory inertia of the viscoelastic layer. Spectral element analysis is conducted to evaluate the accuracy of the spectral element formulated in this paper. The spectral element model of ACLD beam is found to match the measured frequency responses more closely when compared with the finite element model.