Analytical and Experimental Studies on the Large Amplitude Free Vibrations of Variable-Arc-Length Beams

2005 ◽  
Vol 11 (7) ◽  
pp. 923-947 ◽  
Author(s):  
T. Pulngern ◽  
S. Chucheepsakul ◽  
M. W. Halling
Keyword(s):  
Large Amplitude ◽  
Experimental Studies ◽  
Equation Of Motion ◽  
Free Vibrations ◽  
Accurate Solution ◽  
Mode Shapes ◽  
Arc Length ◽  
Linear Eigenvalue Problem ◽  
Non Linear ◽  
Direct Numerical Integration

Using the finite element method, we investigate large amplitude vibrations of horizontal variable-arc-length beams, considering the effect of large initial static sag deflections due to self-weight. The variability in beam arc-length arises from one end being pinned, and the other end being supported by a frictionless roller at a fixed distance from the pinned end. Using Lagrange’s equation of motion, the large amplitude free vibration equation of motion is derived based on the variational formulation. Included in the formulation are the energy dissipation due to large bending using the exact non-linear expression of curvature and the non-linearity arising from axial force. The non-linear eigenvalue problem is solved by the direct iteration method to obtain the beam’s non-linear frequencies and corresponding mode shapes for specified vibration amplitudes. We also present changes in the frequency of vibration as a function of amplitude, demonstrating the beam non-linearity. A more accurate solution analyzed in the frequency domain of the direct numerical integration method is adopted as an alternative solution. Large amplitude vibration experimental modal analysis was also conducted to complement the analytical results. The measured results were found to be in good agreement with those obtained from two analytical solutions.

Non-Linear Vibration of Thick Dielectric Membrane Disks With Radial Loads

2020 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert L. Lowe
Keyword(s):  
Frequency Response ◽  
Applied Voltage ◽  
Large Amplitude ◽  
Equation Of Motion ◽  
Membrane Thickness ◽  
Linear Vibration ◽  
Response Characteristics ◽  
Wide Range ◽  
Non Linear ◽  
Large Stretch

Abstract This study analyzes the large-amplitude, non-linear vibration of dielectric elastomer membrane disks with applied voltages through their thickness and mechanical loads applied radially around their outer circumferential surface. The material is modeled as an isotropic ideal dielectric, with the large-stretch mechanical stiffening captured using the Gent hyperelastic constitutive model. The fully non-linear equation of motion for the coupled electromechanical system is derived using Hamilton’s principle. The disk comes to a steady equilibrium where the compressive stresses due to the applied voltage balance the tensile stresses from the applied radial loads. The equilibria are calculated numerically for a wide range of radial loads, applied voltages, and limiting stretches. It is possible for the disk to have two stable steady equilibria at given radial load and applied voltage, which gives rise to an instability where extreme stretches occur for infinitesimal changes in applied voltage. The equation of motion is determined for small vibrations of the system about equilibrium. Unlike for thin membrane disks, the vibrating mass of thick membrane disks depends on the steady equilibrium stretch. The natural frequency for membrane disks meaningfully decreases with increasing thickness due to the inertia associated with dynamic changes in the membrane thickness. The amount of axial inertia depends on the ratio of the nominal disk thickness to its radius and the steady equilibrium stretch. Large amplitude vibrations are numerically investigated for a wide range of system parameters. The frequency response characteristics of circular membranes due to sinusoidal voltage fluctuations are analyzed about small and large equilibrium stretches. Whereas axial inertia meaningfully alters the frequency response about small equilibrium stretches, it has negligible effects on the frequency response about large equilibrium stretches.

scholarly journals Free Vibrations with Large Amplitude of Axially Loaded Beams on an Elastic Foundation Using the Adomian Modified Decomposition Method

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Desmond Adair ◽  
Askar Ibrayev ◽  
Alima Tazabekova ◽  
Jong R. Kim
Keyword(s):  
Governing Equation ◽  
Large Amplitude ◽  
Decomposition Method ◽  
Natural Frequencies ◽  
Axial Loading ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Transverse Vibrations ◽  
Axially Loaded ◽  
Mathematical Operations

Analytical solutions describing free transverse vibrations with large amplitude of axially loaded Euler–Bernoulli beams for various end restrains resting on a Winkler one-parameter foundation are obtained using the Adomian modified decomposition method (AMDM). The AMDM allows the governing equation to become a recursive algebraic equation, and, after some additional simple mathematical operations, the equations can be cast as an eigenvector problem whose solution results in the calculation of natural frequencies and corresponding closed-form series solution of the mode shapes. Important to the use of the Adomian modified decomposition method is the treatment of the nonlinear Fredholm integral coefficient, which forms part of the governing equation. In addition to the calculation of natural frequencies and mode shapes, investigations are made of the effects on the free vibrations of the Winkler parameter and of increasing the axial loading.

Re-investigation of large-amplitude free vibrations of beams using finite elements

1990 ◽  
Vol 143 (2) ◽  
pp. 351-355 ◽  
Author(s):  
G. Singh ◽  
G. Venkateswara Rao ◽  
N.G.R. Iyengar
Keyword(s):  
Finite Elements ◽  
Large Amplitude ◽  
Free Vibrations

Non-linear torsional oscillation of a two-degree-of-freedom system incorporating a Hooke joint

1964 ◽  
Vol 277 (1368) ◽  
pp. 92-106 ◽  
Keyword(s):  
Large Amplitude ◽  
Critical Speed ◽  
Torsional Oscillation ◽  
Degree Of Freedom ◽  
Non Linear ◽  
Two Degree Of Freedom

The non-linear torsional oscillation of the system is analyzed by means of a variant of Kryloff and Bogoliuboff’s method. It is shown that each mode of the system can perform oscillations of large amplitude in a number of critical speed ranges, and that hysteresis effects and discontinuous jumps in amplitude are to be expected in these speed ranges if the damping is light.

Free Vibrations of Thick, Complete Conical Shells of Revolution From a Three-Dimensional Theory

2005 ◽  
Vol 72 (5) ◽  
pp. 797-800 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Ritz Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Shells Of Revolution ◽  
Dimensional Theory ◽  
Conical Shells ◽  
Cylindrical Coordinate

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution in which the bottom edges are normal to the midsurface of the shells based upon the circular cylindrical coordinate system using the Ritz method. Comparisons are made between the frequencies and the corresponding mode shapes of the conical shells from the authors' former analysis with bottom edges parallel to the axial direction and the present analysis with the edges normal to shell midsurfaces.

scholarly journals Comparison of Two Numerical Algorithms for Computing the Effects of Mistuning of Fan Flutter

2016 ◽  
Author(s):  
L. Salles ◽  
M. Vahdati
Keyword(s):  
Numerical Models ◽  
Three Dimensional ◽  
Numerical Algorithms ◽  
Mode Shapes ◽  
Minimal Amount ◽  
Aeroelastic System ◽  
Cpu Time ◽  
Influence Coefficients ◽  
Non Linear ◽  
The Time Domain

The aim of this paper is to study the effects of mistuning on fan flutter and to compare the prediction of two numerical models of different fidelity. The high fidelity model used here is a three-dimensional, whole assembly, time-accurate, viscous, finite-volume compressible flow solver. The Code used for this purpose is AU3D, written in Imperial College and validated for flutter computations over many years. To the best knowledge of authors, this is the first time such computations have been attempted. This is due to the fact that, such non-linear aeroelastic computations with mistuning require large amount of CPU time and cannot be performed routinely and consequently, faster (low fidelity) models are required for this task. Therefore, the second model used here is the aeroelastic fundamental mistuning model (FMM) and it based on an eigenvalue analysis of the linearized modal aeroelastic system with the aerodynamic matrix calculated from the aerodynamic influence coefficients. The influence coefficients required for this algorithm are obtained from the time domain non-linear Code by shaking one blade in the datum (tuned) frequency and mode. Once the influence coefficients have been obtained, the computations of aero damping require minimal amount of CPU time and many different mistuning patterns can be studied. The objectives of this work are to: 1. Compare the results between the two models and establish the capabilities/limitations of aeroelastic FMM, 2. Check if the introduction of mistuning would bring the experimental and computed flutter boundaries closer, 3. Establish a relationship between mistuning and damping. A rig wide-chord fan blade, typical of modern civil designs, was used as the benchmark geometry for this study. All the flutter analyses carried out in this paper are with frequency mistuning, but the possible consequences of mistuned mode shapes are briefly discussed at the end of this paper. Only the first family of modes (1F, first flap) is considered in this work. For the frequency mistuning analysis, the 1F frequency is varied around the annulus but the 1F mode shapes remain the same for all the blades. For the mode shape mistuning computations, an FE analysis of the whole assembly different mass blades is performed. The results of this work clearly show the importance of mistuning on flutter. It also demonstrates that when using rig test data for aeroelastic validation of CFD codes, the amount mistuning present must be known. Finally, it should be noted that the aim of this paper is the study of mistuning and not steady/unsteady validation of a CFD code and therefore minimal aerodynamic data are presented.

Dynamic Model and Simulation of Flag Vibrations Modeled as a Membrane

2014 ◽  
Author(s):  
Gary Frey ◽  
Ben Carmichael ◽  
Joshua Kavanaugh ◽  
S. Nima Mahmoodi
Keyword(s):  
Constant Velocity ◽  
Cross Flow ◽  
Vertical Direction ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
System Response ◽  
Wind Flow ◽  
Pressure Function ◽  
Model And Simulation ◽  
Reasonable Model

A flag is modeled as a membrane to investigate the two-dimensional characteristics of the vibration response to an uniform wind flow. Both the affecting tension and pressure functions for the wind flow with constant velocity are introduced and utilized in the modeling. In this case, the tension is caused by the weight of the flag. The pressure function is a function describing the pressure variations caused on the flag when in uniform flow. The pressure function is found by assuming that the air flow is relatively slow and that the flag is wide enough to minimize cross flow at the boundaries. An analysis of the downstream motion of the flag is necessary as well. Hamilton’s principle is employed to derive the partial differential equation of motion. The flag is oriented in the vertical direction to neglect the effect of the flag’s weight on the system’s response. Galerkin’s method is used to solve for the first four mode shapes of the system, and the system response is numerically solved. Simulations reveal a very reasonable model when the flag is modeled as a membrane.

Earthquake response of linear-elastic arch-frames using exact curved beam formulations

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Baran Bozyigit
Keyword(s):  
Dynamic Stiffness ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Opening Angle ◽  
Earthquake Response ◽  
Response Analysis ◽  
Curved Beam ◽  
Curved Beams ◽  
Base Shear ◽  
Content Type

PurposeThis study aims to obtain earthquake responses of linear-elastic multi-span arch-frames by using exact curved beam formulations. For this purpose, the dynamic stiffness method (DSM) which uses exact mode shapes is applied to a three-span arch-frame considering axial extensibility, shear deformation and rotational inertia for both columns and curved beams. Using exact free vibration properties obtained from the DSM approach, the arch-frame model is simplified into an equivalent single degree of freedom (SDOF) system to perform earthquake response analysis.Design/methodology/approachThe dynamic stiffness formulations of curved beams for free vibrations are validated by using the experimental data in the literature. The free vibrations of the arch-frame model are investigated for various span lengths, opening angle and column dimensions to observe their effects on the dynamic behaviour. The calculated natural frequencies via the DSM are presented in comparison with the results of the finite element method (FEM). The mode shapes are presented. The earthquake responses are calculated from the modal equation by using Runge-Kutta algorithm.FindingsThe displacement, base shear, acceleration and internal force time-histories that are obtained from the proposed approach are compared to the results of the finite element approach where a very good agreement is observed. For various span length, opening angle and column dimension values, the displacement and base shear time-histories of the arch-frame are presented. The results show that the proposed approach can be used as an effective tool to calculate earthquake responses of frame structures having curved beam elements.Originality/valueThe earthquake response of arch-frames consisting of curved beams and straight columns using exact formulations is obtained for the first time according to the best of the author’s knowledge. The DSM, which uses exact mode shapes and provides accurate free vibration analysis results considering each structural members as one element, is applied. The complicated structural system is simplified into an equivalent SDOF system using exact mode shapes obtained from the DSM and earthquake responses are calculated by solving the modal equation. The proposed approach is an important alternative to classical FEM for earthquake response analysis of frame structures having curved members.

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