Elastodynamic Analysis of a Completely Elastic System

1977 ◽  
Vol 99 (3) ◽  
pp. 604-609 ◽  
Author(s):  
D. Kohli ◽  
D. Hunter ◽  
G. N. Sandor
Keyword(s):  
Vibration Analysis ◽  
Elastic System ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Lagrange Equations ◽  
Elastic Supports ◽  
Linear Ordinary Differential Equations ◽  
Elastic Links ◽  
Inertia Forces ◽  
Crank Mechanism

The completely elastic system considered for this vibration analysis consists of an offset slider-crank mechanism having (a) elastic supports and mountings of the mechanism permitting translational vibrations of the shafts and supports, (b) elastic shafts permitting torsional vibrations, (c) elastic links of the mechanism which deform due to external or internal body forces and allow flexural and axial vibrations. Both the effect of the deformations caused by the inertia forces in the mechanism links, shafts, and supports and the effect of change in the inertia forces due to these deformations are taken into account in constructing a general mathematical model for conducting elastodynamic analysis. The rigid displacements (finite and infinitesimal) of the mechanism links due to deformations in the support are evaluated using a truncated Taylor series approximation. Deformation in the links caused by the inertia forces is approximated by a finite number of terms in a Fourier series using the Raleigh-Ritz method. The Lagrange equations of motion are used to obtain coupled time varying linear ordinary differential equations of motion for the vibration analysis of the slider-crank mechanism. The method in general may be applied to any planar or spatial system consisting of elastic links, elastic shafts, and elastic supports. Numerical examples are presented for illustration.

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.

Dynamic Behavior and Stability of a Rotor Under Base Excitation

2006 ◽  
Vol 128 (5) ◽  
pp. 576-585 ◽  
Author(s):  
M. Duchemin ◽  
A. Berlioz ◽  
G. Ferraris
Keyword(s):  
Dynamic Behavior ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Base Excitation ◽  
Flexible Rotor ◽  
Lagrange Equations ◽  
Rotor Systems ◽  
Simple Systems ◽  
Flexible Rotor Systems

The dynamic behavior of flexible rotor systems subjected to base excitation (support movements) is investigated theoretically and experimentally. The study focuses on behavior in bending near the critical speeds of rotation. A mathematical model is developed to calculate the kinetic energy and the strain energy. The equations of motion are derived using Lagrange equations and the Rayleigh-Ritz method is used to study the basic phenomena on simple systems. Also, the method of multiple scales is applied to study stability when the system mounting is subjected to a sinusoidal rotation. An experimental setup is used to validate the presented results.

scholarly journals Development of a Combined-Ritz Method for Modeling 3D Acoustics in Candu Fuel Sub-Channels

2021 ◽  
Author(s):  
David Villero
Keyword(s):  
Equations Of Motion ◽  
Ritz Method ◽  
Axial Direction ◽  
Acoustic System ◽  
Lagrange Equations ◽  
Acoustic Wave Propagation ◽  
Absorbing Material ◽  
Acoustic Pressure Wave ◽  
System Equations ◽  
Triangular Elements

A combined finite element-Ritz method is developed to effectively model the 3D lowfrequency acoustics in CANDU fuel sub-channels. The complex acoustic behavior of CANDU fuel sub-channels in the cross section is captured using the six-node isoparametric triangular elements; and the acoustic wave propagation in the axial direction is modeled using the polynomials of order n. The Lagrange equations are utilized to formulate the system equations of motion. The acoustic system considered in this study consists of pipe-like medium (water) with rigid and smooth walls. At the inlet of the fuel channel acoustic system, an acoustic pressure wave is prescribed to simulate the pulsation induced by the main feeder pumps. At the outlet, the acoustic system is assumed to interact with a reacting and absorbing material with prescribed acoustic impedance. The method was tested for several scenarios of interest. Numerical results obtained are in excellent agreement with the analytical and ANSYS solutions.

scholarly journals Minimization of dynamic joint reaction forces of the 2-DOF serial manipulators based on interpolating polynomials and counterweights

2015 ◽  
Vol 42 (4) ◽  
pp. 249-260 ◽  
Author(s):  
Slavisa Salinic ◽  
Marina Boskovic ◽  
Radovan Bulatovic
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Lagrange Equations ◽  
Symbolic Form ◽  
Serial Manipulators ◽  
Joint Reaction Forces ◽  
Reaction Forces ◽  
Inertia Forces ◽  
Joint Reaction ◽  
Selection Of

This paper presents two ways for the minimization of joint reaction forces due to inertia forces (dynamic joint reaction forces) in a two degrees of freedom (2-DOF) planar serial manipulator. The first way is based on the optimal selection of the angular rotations laws of the manipulator links and the second one is by attaching counterweights to the manipulator links. The influence of the payload carrying by the manipulator on the dynamic joint reaction forces is also considered. The expressions for the joint reaction forces are obtained in a symbolic form by means of the Lagrange equations of motion. The inertial properties of the manipulator links are represented by dynamical equivalent systems of two point masses. The weighted sum of the root mean squares of the magnitudes of the dynamic joint reactions is used as an objective function. The effectiveness of the two ways mentioned is discussed.

Vibration Analysis of Thin Plates Subject to Piezoelectric Actuation: A New Perspective in Modeling and Numerical Analysis

2011 ◽  
Author(s):  
Parikshit Mehta ◽  
Nader Jalili
Keyword(s):  
Piezoelectric Actuator ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Model Development ◽  
Thin Plates ◽  
Shape Functions ◽  
Assumed Mode Method ◽  
New Perspective ◽  
Forced Vibration Analysis

This paper undertakes model development and numerical simulations of vibration problem of piezoelectrically actuated thin plates with a holistic perspective. Constitutive laws governing piezoelectric actuator are integrated with the potential and kinetic energies of combined plate-actuator system. The equations of motions are derived using variational approach and verified with results obtained by Newton’s equilibrium approach. It is verified that the field coupled components associated with piezoelectric actuator appear as distributed moments over the area of the actuator. The equations of motion are solved using modal analysis deploying Raleigh Ritz method utilizing Boundary Characteristic Orthogonal Polynomials (BCOP). The shape functions generated using this method is used in Assumed Mode Method (AMM) to numerically simulate forced vibration analysis. Since Raleigh Ritz analysis with BCOP can be deployed with the plates of all the geometries, minor modifications in selecting the shape functions enables one to use the same method to calculate natural frequencies of annular plate as well.

Free Vibration Analysis of Planar Flexible Mechanisms

2003 ◽  
Vol 125 (4) ◽  
pp. 764-772 ◽  
Author(s):  
S. D. Yu ◽  
F. Xi
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Summation Method ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Modal Summation ◽  
Gyroscopic Effects ◽  
Flexible Mechanisms

This paper presents a methodology for accurate free vibration analysis of planar flexible mechanisms. Each flexible body is considered as a beam and modelled using higher-order beam elements for longitudinal and flexural deformations. The global equations of motion for a mechanism consisting of multiple flexible bodies are formulated using the augmented Lagrange equations. Free vibration analyses are conducted at desired fast Fourier configurations to determine instantaneous structural natural frequencies and structural mode shapes. Dynamical frequencies and dynamical mode shapes incorporating the gyroscopic effects and dynamic axial loads are obtained using the modal summation method. Numerical results and comparisons are given for a rotating beam and two four-bar crank-rocker mechanisms.

scholarly journals Development of a Combined-Ritz Method for Modeling 3D Acoustics in Candu Fuel Sub-Channels

2021 ◽  
Author(s):  
David Villero
Keyword(s):  
Equations Of Motion ◽  
Ritz Method ◽  
Axial Direction ◽  
Acoustic System ◽  
Lagrange Equations ◽  
Acoustic Wave Propagation ◽  
Absorbing Material ◽  
Acoustic Pressure Wave ◽  
System Equations ◽  
Triangular Elements

A combined finite element-Ritz method is developed to effectively model the 3D lowfrequency acoustics in CANDU fuel sub-channels. The complex acoustic behavior of CANDU fuel sub-channels in the cross section is captured using the six-node isoparametric triangular elements; and the acoustic wave propagation in the axial direction is modeled using the polynomials of order n. The Lagrange equations are utilized to formulate the system equations of motion. The acoustic system considered in this study consists of pipe-like medium (water) with rigid and smooth walls. At the inlet of the fuel channel acoustic system, an acoustic pressure wave is prescribed to simulate the pulsation induced by the main feeder pumps. At the outlet, the acoustic system is assumed to interact with a reacting and absorbing material with prescribed acoustic impedance. The method was tested for several scenarios of interest. Numerical results obtained are in excellent agreement with the analytical and ANSYS solutions.

scholarly journals A least action principle for interceptive walking

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Soon Ho Kim ◽  
Jong Won Kim ◽  
Hyun Chae Chung ◽  
MooYoung Choi
Keyword(s):  
Social Systems ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Action Principle ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Least Action ◽  
The Least Action Principle ◽  
Principle Of Least Action ◽  
Human Participants

AbstractThe principle of least effort has been widely used to explain phenomena related to human behavior ranging from topics in language to those in social systems. It has precedence in the principle of least action from the Lagrangian formulation of classical mechanics. In this study, we present a model for interceptive human walking based on the least action principle. Taking inspiration from Lagrangian mechanics, a Lagrangian is defined as effort minus security, with two different specific mathematical forms. The resulting Euler–Lagrange equations are then solved to obtain the equations of motion. The model is validated using experimental data from a virtual reality crossing simulation with human participants. We thus conclude that the least action principle provides a useful tool in the study of interceptive walking.

scholarly journals A Unified Spectro-Geometric-Ritz Method for Vibration Analysis of Open and Closed Shells with Arbitrary Boundary Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-30 ◽  
Author(s):  
Dongyan Shi ◽  
Yunke Zhao ◽  
Qingshan Wang ◽  
Xiaoyan Teng ◽  
Fuzhen Pang
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Analysis Model ◽  
Arbitrary Boundary ◽  
Fourier Cosine ◽  
Auxiliary Functions ◽  
Arbitrary Boundary Conditions ◽  
Closed Shells

This paper presents free vibration analysis of open and closed shells with arbitrary boundary conditions using a spectro-geometric-Ritz method. In this method, regardless of the boundary conditions, each of the displacement components of open and closed shells is represented simultaneously as a standard Fourier cosine series and several auxiliary functions. The auxiliary functions are introduced to accelerate the convergence of the series expansion and eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries. The boundary conditions are modeled using the spring stiffness technique. All the expansion coefficients are treated equally and independently as the generalized coordinates and determined using Rayleigh-Ritz method. By using this method, a unified vibration analysis model for the open and closed shells with arbitrary boundary conditions can be established without the need of changing either the equations of motion or the expression of the displacement components. The reliability and accuracy of the proposed method are validated with the FEM results and those from the literature.

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