Modal Superposition Approach Research Articles

VIBRATIONS OF SHALLOW SHELLS DUE TO REMOVAL OF FORMWORK

This paper is about the forced vibrations of thin-walled shallow shells having rectangular platform and constant thickness and curvature. Assumptions of linear behaviour in both material and kinematic properties of the system are assumed, and an analytical solution is obtained for natural frequencies and mode shapes. The forced vibrations of the shell are computed using the modal superposition approach. The formulation is applied to solve the transient response of a reinforced concrete shallow shell due to removal of the framework.The influence of structural defects is included in an approximate way to estimate forced vibrations in a deteriorated shell. Parametric studies indicate the influence of the main parameters affecting the behaviour of the shell.

Improved fixed interface method for modal synthesis

The paper takes a fresh look at the important topic of substructure synthesis. An improved technique for synthesizing the modes obtained with fixed interface coordinates is proposed. The reduction in the size of a substructure model is achieved through a higher order mode superposition approach. For the first time, an analytical rationale for the popular Craig-Bampton-Hurty method is presented. It is shown that the CraigBampton-Hurty method is a special case of the proposed method. The proposed approach can be used to improve the accuracy of the calculated eigenproperties by utilizing the higher order component modes but without calculating additional normal modes of the substructures. Numerical results are presented to demonstrate the effectiveness of the proposed method with the increasing order of the mode superposition approach used. The approach is expected to be efficient if the degrees of freedom at the interface of the substructures are relatively small compared with the degrees of freedom of the substructures.

A method of non-linear modal identification from frequency response tests

A theoretical and experimental study of non-linear systems governed by Duffing's equation is presented in this study. The non-linear mode concept is applied to analyze the steady state responses of non-linear systems driven by a harmonic force. In the experimental context, the present modal identification method permits one to obtain the non-linear modal parameters from frequency response tests. By comparison with other methods, the proposed methods appear very interesting in regard to computational time, formulation, and they necessitate fewer computer resources. An experimental work on the steady state response of a beam with local non-linear stiffness is reported to justify the methods. Many aspects of theoretical results, such as the dependence of resonance frequency on the amplitude of vibration and the downward and upward jump phenomena are observed experimentally. The experimental results are in good agreement with theoretical predictions based upon the non-linear modal superposition approach.

Forced vibration response of a twisted non-uniform rotating blade

The in-plane (lag) and out-of-plane (flap) dynamic deflections of a flexible twisted non-uniform rotating blade subjected to externally applied arbitrary dynamic loading are determined through a mode superposition approach. The procedure utilizes an integrating matrix method to determine the eigen-properties of the system. A technique which avoids the need for a numerical integration of the modal equations by step-by-step or other approaches is incorporated into the resulting computationally efficient code.

Frequency response smoothing, matrix assembly and structural paths: A new approach for structural dynamics up to high frequencies

Frequency response smoothing by geometric mean values, in conjunction with a solution written in terms of structural paths, provides a new approach for estimating the mean frequency responses of complex structures subjected to mechanical excitation from low to high frequencies. A theoretical formulation is derived for continuous structures with discrete connections between components. This formulation is based on the assembly of the smoothed dynamic stiffnesses of each component in a classical way, followed by a manipulation in terms of structural paths to provide the smoothed frequency responses of the whole structure. The results can be easily interpreted due to the physical meaning of the structural paths. This approach is applied to trusses for illustration; however, it can be extended to any structure provided that the formulation of each component is established, as in a finite element approach. Computer time efficiency for large structures will depend on the selection of the “shortest” paths (similar to modal truncation for modal superposition approach). Finally, the application to dynamic substructuring with component dynamic stiffnesses or flexibilities is presented, showing promising capabilities for analysis and testing of complex structures up to high frequencies.

A comparison of harmonic acceleration method with the other commonly used methods for calculation of dynamic transient response

Comparison of the central finite difference method, the Houbolt method, the Wilson θ method, the Newmark method and the harmonic acceleration method is carried out by the modal superposition approach. Amplitude decay, period elongation and spectral radius are adopted as a basis of comparison. An extension to evaluation of load operator error is proposed. Among the methods considered, only integration of modal equations by the harmonic acceleration method has no numerical damping, amplitude decay or period elongation. Several instructive examples are presented.

Nonlinear Dynamic Analysis of Guyed Tower Platforms

A method of analysis is presented to predict the nonlinear dynamic behavior of compliant offshore structures, in particular guyed tower platforms. The structural model is analyzed in time domain using the normal mode superposition approach. Nonlinearities due to fluid-structure interaction are considered. The nonlinear stiffness of the mooring system is included by suitable modification of the forcing function. Secondary overturning effects due to large deflection are introduced by equivalent lateral forces at each mass point. In analogy to the treatment of the nonlinear mooring stiffness, the nonlinearity of the soil-pile foundation can be accounted for by applying external forces and moments. By way of example, the results of dynamic analyses performed on a guyed tower designed for the North Sea are included.

Seismogram synthesis using normal mode superposition: the locked mode approximation

Summary. A normal mode superposition approach is used to synthesize complete seismic codas for flat layered earth models and the P-SV phases. Only modes which have real eigenwavenumbers are used so that the search for eigenvalues in the complex wavenumber plane is confined to the real axis. In order to synthesize early P-wave arrivals by summing a number of‘trapped’modes, an anomalously high velocity cap layer is added to the bottom of the structure so that most of the seismic energy is contained in the upper layers as high-order surface waves. Causality arguments are used to define time windows for which the resulting synthetic seismograms are close approximations to the exact solutions without the cap layer. The traditional Thomson—Haskell matrix approach to computing the normal modes is reformulated so that numerical problems encountered at high frequencies are avoided and numerical results of the locked mode approximation are given.

On the dynamic spatial response of a heat exchanger tube with intermittent baffle contacts

Flow-induced vibration in heat exchanger tubes can result in fretting wear at the baffle supports and subsequent tube failure. As one step in correlating the random flow excitation to the rate of fretting wear, this paper presents a dynamic finite element technique for predicting the motions and baffle contact forces of a single heat exchanger tube. Using a modal superposition approach, the modal equations of motion are generated and numerically integrated. The predicted results are compared with experimental data for both planar and spatial vibration of harmonically-excited cantilevered beams with a clearance support at the free end.

Sloshing of Liquids in Circular Canals and Spherical Tanks

Abstract : Theoretical calculations are made of the natural modes and frequencies of small-amplitude sloshing of liquids in partially filled circular canals and spherical tanks. An integral-equation approach is used to analyze the circular canal for arbitrary depth of liquid. A similar approach for the spherical tank provides modes and frequencies for the nearly-full and half-full cases. These results, together with the known behavior of the nearly- empty tank, are used in conjunction with the trends established for the circular canal as a basis for estimating frequencies for arbitrary depth of liquid in the spherical tank. The dynamic analysis of the container-fluid system by means of the mode-superposition approach is discussed, and modal parameters required in such analyses are presented.