scholarly journals Nonlinear Responses of Inextensible Cantilever and Free–Free Beams Undergoing Large Deflections

2018 ◽  
Vol 85 (5) ◽  
Author(s):  
Kevin McHugh ◽  
Earl Dowell
Keyword(s):  
Equations Of Motion ◽  
Ritz Method ◽  
Large Deflections ◽  
Constraint Force ◽  
Nonlinear Responses ◽  
Nonlinear Motion ◽  
Resonant Modes ◽  
Cantilevered Beam ◽  
Point Forces ◽  
Free Beam

A theoretical and computational model has been developed for the nonlinear motion of an inextensible beam undergoing large deflections for cantilevered and free–free boundary conditions. The inextensibility condition was enforced through a Lagrange multiplier which acted as a constraint force. The Rayleigh–Ritz method was used by expanding the deflections and the constraint force in modal series. Lagrange's equations were used to derive the equations of motion of the system, and a fourth-order Runge–Kutta solver was used to solve them. Comparisons for the cantilevered beam were drawn to experimental and computational results previously published and show good agreement for responses to both static and dynamic point forces. Some physical insights into the cantilevered beam response at the first and second resonant modes were obtained. The free–free beam condition was investigated at the first and third resonant modes, and the nonlinearity (primarily inertia) was shown to shift the resonant frequency significantly from the linear natural frequency and lead to hysteresis in both modes.

Nonlinear Responses of Inextensible Cantilever and Free-Free Beams Undergoing Large Deflections

2017 ◽  
Author(s):  
Kevin McHugh ◽  
Earl Dowell
Keyword(s):  
Equations Of Motion ◽  
Ritz Method ◽  
Large Deflections ◽  
Constraint Force ◽  
Nonlinear Responses ◽  
Nonlinear Motion ◽  
Resonant Modes ◽  
Cantilevered Beam ◽  
Point Forces ◽  
Free Beam

A theoretical and computational model has been developed for the nonlinear motion of an inextensible beam undergoing large deflections for cantilevered and free-free boundary conditions. The inextensibility condition was enforced through a Lagrange multiplier which acted as a constraint force. The Rayleigh-Ritz method was used by expanding the deflections and the constraint force in modal series. Lagrange’s Equations were used to derive the equations of motion of the system, and a 4th order Runge-Kutta solver was used to solve them. Comparisons for the cantilevered beam were drawn to experimental and computational results previously published and show good agreement for responses to both static and dynamic point forces. Some physical insights into the cantilevered beam response at the 1st and 2nd resonant modes were obtained. The free-free beam condition was investigated at the 1st and 3rd resonant modes and the nonlinearity (primarily inertia) was shown to shift the resonant frequency significantly from the linear natural frequency and lead to hysteresis in both modes.

Equations of Motion for an Inextensible Beam Undergoing Large Deflections

2016 ◽  
Vol 83 (5) ◽  
Author(s):  
Earl Dowell ◽  
Kevin McHugh
Keyword(s):  
Boundary Conditions ◽  
Lagrange Multiplier ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Large Deflections ◽  
Lagrange Equations ◽  
Cantilevered Beam ◽  
Alternative Approaches ◽  
Transverse Deflection ◽  
Free Beam

The Euler–Lagrange equations and the associated boundary conditions have been derived for an inextensible beam undergoing large deflections. The inextensibility constraint between axial and transverse deflection is considered via two alternative approaches based upon Hamilton's principle, which have been proved to yield equivalent results. In one approach, the constraint has been appended to the system Lagrangian via a Lagrange multiplier, while in the other approach the axial deflection has been expressed in terms of the transverse deflection, and the equation of motion for the transverse deflection has been determined directly. Boundary conditions for a cantilevered beam and a free–free beam have been considered and allow for explicit results for each system's equations of motion. Finally, the Lagrange multiplier approach has been extended to equations of motion of cantilevered and free–free plates.

scholarly journals Comparison of Galerkin, POD and Nonlinear-Normal-Modes Models for Nonlinear Vibrations of Circular Cylindrical Shells

2006 ◽  
Author(s):  
M. Amabili ◽  
C. Touze´ ◽  
O. Thomas
Keyword(s):  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Harmonic Excitation ◽  
Nonlinear Responses ◽  
Nonlinear Normal ◽  
The Galerkin Method ◽  
Proper Orthogonal

The aim of the present paper is to compare two different methods available to reduce the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD) and an asymptotic approximation of the Nonlinear Normal Modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the Partial Differential Equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed.

Dynamic analysis of three-layer cylindrical shells with fractional viscoelastic core and functionally graded face layers

2020 ◽  
pp. 107754632096622
Author(s):  
Meisam Shakouri ◽  
Mohammad Reza Permoon ◽  
Abdolreza Askarian ◽  
Hassan Haddadpour
Keyword(s):  
Natural Frequency ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Core Layer ◽  
Functionally Graded ◽  
Viscoelastic Layer ◽  
Viscoelastic Core ◽  
Graded Layers

Natural frequency and damping behavior of three-layer cylindrical shells with a viscoelastic core layer and functionally graded face layers are studied in this article. Using functionally graded face layers can reduce the stress discontinuity in the face–core interface that causes a catastrophic failure in sandwich structures. The viscoelastic layer is expressed using a fractional-order model, and the functionally graded layers are defined by a power law function. Assuming the classical shell theory for functionally graded layers and the first-order shear deformation theory for the viscoelastic core, equations of motion are derived using Lagrange’s equation and then solved via Rayleigh–Ritz method. The obtained results are validated with those in the literature, and finally, the effects of some geometrical and material parameters such as length-to-radius ratio, functionally graded properties, radius and thickness of viscoelastic layer on the natural frequency, and loss factor of the system are considered, and some conclusions are drawn.

Dynamical Equations With Non-Ideal Constraints: New and Old Alternatives

2007 ◽  
Author(s):  
Arun K. Banerjee ◽  
Mark Lemak
Keyword(s):  
Equations Of Motion ◽  
Alternative Form ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Constraint Force ◽  
Kane’S Method ◽  
Kane's Method ◽  
Ellipsoidal Surface ◽  
Ideal Constraint ◽  
The Ideal

This paper deals with the motion of mechanical systems with non-ideal constraints, defined as constraints where the forces associated with the constraint do work. The first objective of the paper is to show that two newly published formulations of equations of motion of systems with such non-ideal constraints are unnecessarily complex for situations where the non-ideal constraint force does not depend on the ideal constraint force, because they introduce and then eliminate these non-working constraint forces. We point out that a method already exists for nonideal constraints, namely, Kane’s equations, which are simpler because, among other things, they are based on automatic elimination of non-working constraints. The examples considered in these recent publications are worked out with Kane’s method to show the applicability and simplicity of Kane’s method for non-ideal constraints. A second objective of the paper is to present an alternative form of equations for systems where the non-ideal constraint force depends on the ideal constraint force, as in the case of Coulomb friction. The formulation is shown to lend itself naturally to also analyzing impact dynamics. The method is applied to the dynamics of a slug moving against friction on a moving ellipsoidal surface. Such a crude model may simulate, in essence, propellant motion in a tank in zero-g, or during docking of a spacecraft.

Multibody Dynamics Incorporating Deployment of Flexible Structures

1996 ◽  
Vol 118 (2) ◽  
pp. 237-241 ◽  
Author(s):  
S. S. K. Tadikonda ◽  
R. P. Singh ◽  
S. Stornelli
Keyword(s):  
Multibody Dynamics ◽  
Solution Method ◽  
Equations Of Motion ◽  
Flexible Structures ◽  
Open Loop ◽  
Space Structures ◽  
Shape Functions ◽  
Recursive Solution ◽  
Multi Body ◽  
Free Beam

The equations of motion for a flexible structure during deployment from and retraction into a base that is part of an open-loop multi-body chain are derived. The eigenfunctions of a fixed-free beam are used as the shape functions and their properties are exploited to express various domain integral terms as explicit functions of the instantaneous deployed length. The essential contributions of the present paper are the modeling of flexible body deployment with mass transfer and a recursive solution method for the dynamics. The deployment or retraction of space structures such as the SAFE Extension Mast can be simulated using this model. The model is presented in a format that is suitable for implementation in multibody dynamics codes.

Equations of Nonlinear Motion of Viscoelastic Beams

2005 ◽  
Author(s):  
S. Nima Mahmoodi ◽  
Siamak E. Khadem ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Equations Of Motion ◽  
Elastic Beam ◽  
Longitudinal Direction ◽  
Nonlinear Damping ◽  
Equation Of Motion ◽  
Bending Vibration ◽  
Viscoelastic System ◽  
Nonlinear Beam ◽  
Nonlinear Motion ◽  
Large Amplitude Vibrations

A viscoelastic nonlinear beam with cubic nonlinearities is considered. In order to obtain the equations of nonlinear motion of the beam for large deformation vibrations, the Lagrangian dynamics and Hamilton principle is used. It is considered that the beam vibrates in two directions, one in longitudinal direction and the other in the transverse direction. Large amplitude vibrations cause the nonlinearities in inertia and geometry terms. Also, due to viscoelastic property of the beam, a nonlinear damping term is appeared in the equations of motion. Using the condition of inextensible beams, the equation of motion and boundary conditions of bending vibration of a Kelvin-Voigt viscoelastic beam has been obtained. Finally, if one considers the damping coefficient to be equal to zero in the obtained equation of motion of viscoelastic system then, an equation of motion for the elastic beam will be obtained.

Ritz Series Analysis of Rotating Shaft System: Validation, Convergence, Mode Functions, and Unbalance Response

2002 ◽  
Vol 124 (4) ◽  
pp. 492-501 ◽  
Author(s):  
Nicole L. Zirkelback ◽  
Jerry H. Ginsberg
Keyword(s):  
Equations Of Motion ◽  
Ritz Method ◽  
Natural Mode ◽  
Rotating Shaft ◽  
Disk System ◽  
Element Analysis ◽  
Cross Sectional ◽  
Energy Functionals ◽  
Unbalance Response ◽  
Gyroscopic Coupling

A shaft with attached rigid disks is modeled as a rotating Timoshenko beam supported by nonconservative, flexible bearing supports. The continuous shaft-disk system is described with kinetic and potential energy functionals that fully account for transverse shear, translational and rotatory inertia, and gyroscopic coupling. Ritz series expansions are used to describe the flexural displacements and cross-sectional rotations about orthogonal fixed axes. The equations of motion are derived from Lagrange’s equations and placed in a state-space form that preserves the skew-symmetric gyroscopic matrix as well as the full effects of the bearings. Both the general and adjoint eigenproblems for the nonsymmetric equations are solved. Bi-orthogonality conditions lead to the ability to evaluate dynamic response via modal analysis. Whirl speeds and logarithmic decrements calculated with the present model are verified with a finite element analysis. The present work provides two ways of evaluating the convergence of results to demonstrate an advantage of the Ritz method over other discretization methods. Natural mode functions and unbalance response are calculated for an example system.

scholarly journals Asymmetric Vibrations and Chaos in Spherical Caps under Uniform Time-varying Pressure Fields

2021 ◽  
Author(s):  
Giovanni Iarriccio ◽  
Antonio Zippo ◽  
Francesco Pellicano
Keyword(s):  
Equations Of Motion ◽  
Ritz Method ◽  
Excitation Frequency ◽  
Lagrange Equations ◽  
Reduced Order ◽  
Pressure Fields ◽  
Time Histories ◽  
Implicit Solver ◽  
Spherical Caps ◽  
External Excitation Frequency

Abstract This paper presents a study on nonlinear asymmetric vibrations in shallow spherical caps under pressure loading. The Novozhilov’s nonlinear shell theory is used for modelling the structural strains. A reduced-order model is developed through the Rayleigh-Ritz method and Lagrange equations. The equations of motion are numerically integrated using an implicit solver. The bifurcation scenario is addressed by varying the external excitation frequency. The occurrence of asymmetric vibrations related to quasi-periodic and chaotic motion is shown through the analysis of time histories, spectra, Poincaré maps, and phase planes.

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