scholarly journals Transient Response of Three-Layered Rings

1977 ◽  
Vol 44 (2) ◽  
pp. 299-304 ◽  
Author(s):  
M. J. Sagartz
Keyword(s):  
Transient Response ◽  
Theoretical Calculations ◽  
Equations Of Motion ◽  
Radial Strain ◽  
Middle Layer ◽  
Computational Technique ◽  
Linear Elastic ◽  
Time Histories ◽  
Measured Strain ◽  
Low Modulus

Hamilton’s principle is used to derive equations of motion for a linear elastic three-layered ring. The theory includes the effects of shear deformation and rotatory inertia in each layer and radial strain effects in the middle layer. A convenient computational technique is developed for transient response evaluation. A companion experimental study was conducted using two different rings. Both rings had aluminum inner and outer layers, but each had a different low-modulus middle layer. Radial impulse loads distributed as a cosine over half the ring circumference, were applied to the outer ring surface, and the transient response was monitored with strain gages mounted on the aluminum layers. Measured strain-time histories were compared with theoretical calculations, and good agreement was obtained.

scholarly journals IV. On the dynamical theory of incompressible viscous fluids and the determination of the criterion

1895 ◽  
Vol 186 ◽  
pp. 123-164 ◽  
Keyword(s):  
Singular Solution ◽  
Physical State ◽  
Theoretical Calculations ◽  
Equations Of Motion ◽  
Dynamical Theory ◽  
Viscous Fluids ◽  
Linear Functions ◽  
Incompressible Viscous Fluids ◽  
Theoretical Results

1. The equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier seems to have first introduced in 1822, and which were much studied by Cauchy and Poisson) were finally shown by St. Venant and Sir Gabriel Stokes, in 1845, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion, with a co-efficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion, Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations so long as the tubes were small and the velocities slow. On the other hand, these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies which rendered the actual motion other than that to which the singular solution referred and not as disproving the assumption.

Vibrations of beams with a breathing crack and large amplitude displacements

2015 ◽  
Vol 230 (1) ◽  
pp. 34-54 ◽  
Author(s):  
Gonçalo Neves Carneiro ◽  
Pedro Ribeiro
Keyword(s):  
Time Domain ◽  
Equations Of Motion ◽  
Shape Functions ◽  
Breathing Crack ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Time Histories ◽  
Linear Effects ◽  
Fourier Spectra ◽  
The Time Domain

The vibrations of beams with a breathing crack are investigated taking into account geometrical non-linear effects. The crack is modeled via a function that reduces the stiffness, as proposed by Christides and Barr (One-dimensional theory of cracked Bernoulli–Euler beams. Int J Mech Sci 1984). The bilinear behavior due to the crack closing and opening is considered. The equations of motion are obtained via a p-version finite element method, with shape functions recently proposed, which are adequate for problems with abrupt localised variations. To analyse the dynamics of cracked beams, the equations of motion are solved in the time domain, via Newmark's method, and the ensuing displacements, velocities and accelerations are examined. For that purpose, time histories, projections of trajectories on phase planes, and Fourier spectra are obtained. It is verified that the breathing crack introduce asymmetries in the response, and that velocities and accelerations can be more affected than displacements by the breathing crack.

Non-Linear Vibrations of Plates Under Thermal and Mechanical Loads

2005 ◽  
Author(s):  
P. Ribeiro
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Implicit Time ◽  
Linear Elastic ◽  
Time Integration Method ◽  
Thermal Fields ◽  
Non Linear ◽  
Linear Vibrations ◽  
The Time Domain ◽  
Constitutive Material

The geometrically non-linear vibrations of plates under the combined effect of thermal fields and mechanical excitations are analyzed. With this purpose, an accurate model based on a p-version, hierarchical, first-order shear deformation finite element is employed. The constitutive material of the plates is linear elastic and isotropic. The equations of motion are solved in the time domain by an implicit time integration method. The temperature and the amplitude of the mechanical excitation are varied, and transitions from periodic to non-periodic motions are found.

Damage Detection in Rods Using Wave Propagation and Regression Analysis

1999 ◽  
Author(s):  
K. T. Feroz ◽  
S. O. Oyadiji
Keyword(s):  
Wave Propagation ◽  
Regression Analysis ◽  
Damage Detection ◽  
Numerical Study ◽  
Time History ◽  
Ball Impact ◽  
Spherical Ball ◽  
Time Histories ◽  
Measured Strain ◽  
Force Time

Abstract The phenomena of wave propagation in rods was studied both numerically and experimentally. The finite element (FE) code ABAQUS was used for the numerical study while PZT (lead zirconium titanate) sensors and a 50 MHz transient recorder were used experimentally to monitor and to capture the propagation of stress pulses. For the study of damage detection in the rods the analyses and the experiments were repeated by introducing slots in a fixed axial location of the rod. A longitudinal wave was induced in the rod via collinear impact which was modelled in the FE analyses using the force-time history computed from the classical Hertz contact theory. In the experimental measurements this was achieved by a spherical ball impact at one plane end of the rods. It is shown that the predicted and measured strain-time histories for the defect-free rod and for the rods with defect correlate quite well. These results also show that defects can be located using the wave propagation phenomena. A regression analysis technique of the predicted and measured strain histories of the defect free rod and of the rod with defect was also performed. The results show that this technique is more efficient for smaller defects. In particular, it is shown that the area enclosed by the regression curve increases as the defect size increases.

scholarly journals Linear Vibration Analysis of Shells Using a Seven-Parameter Spectral/hp Finite Element Model

2020 ◽  
Vol 10 (15) ◽  
pp. 5102
Author(s):  
Carlos Valencia Murillo ◽  
Miguel Gutierrez Rivera ◽  
Junuthula N. Reddy
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Variable Thickness ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Element Model ◽  
Linear Elastic ◽  
Commercial Code ◽  
The Finite Element Model ◽  
Hp Finite Element

In this paper, a seven-parameter spectral/hp finite element model to obtain natural frequencies in shell type structures is presented. This model accounts for constant and variable thickness of shell structures. The finite element model is based on a Higher-order Shear Deformation Theory, and the equations of motion are obtained by means of Hamilton’s principle. Analysis is performed for isotropic linear elastic shells. A validation of the formulation is made by comparing the present results with those reported in the literature and with simulations in the commercial code ANSYS. Finally, results for shell like structures with variable thickness are presented, and their behavior for different ratios r/h and L/r is studied.

Propagation of a nonlinear seismic pulse in an anelastic homogeneous medium

Geophysics ◽  
1993 ◽  
Vol 58 (7) ◽  
pp. 949-963 ◽  
Author(s):  
Dan W. Kosik
Keyword(s):  
Wave Propagation ◽  
Equations Of Motion ◽  
Surrounding Medium ◽  
Homogeneous Medium ◽  
Seismic Surveys ◽  
Linear Elastic ◽  
Cylindrically Symmetric ◽  
Unconsolidated Material ◽  
Linear Pulse ◽  
Implicit Finite Difference

Seismic surveys are often conducted using dynamite charges buried near the surface in unconsolidated material. In such material a large zone near the source should exist wherein nonlinear anelastic wave propagation, can be expected to take place, and have a significant impact on the way in which a seismic pulse forms and how its energy gets distributed into the surrounding medium. To obtain a solution for a propagating pulse in this zone, the equations of motion for nonlinear anelastic wave propagation, good to second order in the displacements, are solved numerically for the problem of a Gaussian pressure pulse acting on the interior cavity of a cylindrically symmetric hole in the medium. An implicit finite‐difference algorithm is used for the solution to the equations of motion for this problem. The anelastic medium is characterized by multivalued stress‐strain relations that exhibit hysteresis, and therefore a loss of energy per cycle, corresponding to a medium with a constant Q factor. Several numerical examples are calculated contrasting the nonlinear anelastic, linear anelastic, and linear elastic propagating pulses to one another. The nonlinear anelastic propagating pulse is found to have an amplitude that is several times larger than would be expected for a pulse in a linear medium and has a peak propagation velocity that is slightly less than that for a linear pulse. Dispersive effects are also evident for the nonlinear pulse.

Unilateral Impact and Contact of Elastic Structures Using Lagrangian Multipliers

2011 ◽  
Author(s):  
Sebastian Tatzko
Keyword(s):  
Equations Of Motion ◽  
Structural Equations ◽  
Contact Forces ◽  
Integration Scheme ◽  
Lagrangian Multiplier ◽  
Elastic Structures ◽  
Lagrangian Multipliers ◽  
Linear Elastic ◽  
Time Stepping ◽  
Numerical Effort

This paper deals with linear elastic structures exposed to impact and contact phenomena. Within a time stepping integration scheme contact forces are computed with a Lagrangian multiplier approach. The main focus is turned on a simplified solving method of the linear complementarity problem for the frictionless contact. Numerical effort is reduced by applying a Craig-Bampton transformation to the structural equations of motion.

The Use of Normal Modes in Problems of Forced Vibration and Impact

1957 ◽  
Vol 61 (560) ◽  
pp. 552-559
Author(s):  
R. P. N. Jones
Keyword(s):  
Forced Vibration ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Very High Frequency ◽  
Linear Elastic ◽  
Modes Of Vibration ◽  
Fundamental Equations ◽  
Very High ◽  
Simple Exposition

SummaryA simple exposition, using d'Alembert's principle and methods of virtual work, is given of the properties and applications of the normal modes of vibration of a linear elastic system. The use of the normal modes in problems of free and forced vibration and dynamic loading is discussed with the aid of simple examples, and it is shown that by these methods dynamical problems for any linear system may be solved without the use of the fundamental equations of motion, provided the natural frequencies and modes of the system are known. In most problems the solutions converge rapidly, so that only the first few modes of vibration need be considered, and in these cases the solution may be modified to give further improvement in convergence. Unsatisfactory convergence may be obtained, however, in problems where there is an exciting force of very high frequency, or an impact of short duration. An approximate allowance may be made for damping, provided this is small.

An Efficient Method for Predicting the Vibratory Response of Linear Structures With Friction Interfaces

1986 ◽  
Vol 108 (4) ◽  
pp. 633-640 ◽  
Author(s):  
E. Bazan ◽  
J. Bielak ◽  
J. H. Griffin
Keyword(s):  
Equations Of Motion ◽  
Solution Procedure ◽  
System Response ◽  
Steady State Response ◽  
Friction Forces ◽  
Practical Applications ◽  
Linear Elastic ◽  
State Response ◽  
Vibratory Response ◽  
Parametric Studies

A simple methodology to study the steady-state response of systems consisting of linear elastic substructures connected by friction interfaces is presented. Assuming that only the first Fourier components of the friction forces contribute significantly to the system response, the differential equations of motion are transformed into a system of algebraic complex equations. Then, an efficient linearized procedure to solve these equations for different normal loads in the friction interfaces is developed. As part of the solution procedure, a criterion to determine the slip-to-stuck transitions in the joint is proposed. Within the assumption that the response is harmonic, any desired accuracy can be obtained with this methodology. Selected numerical examples are presented to illustrate practical applications and the relevant features of the methodology. Due to its simplicity, this methodology is particularly appropriate for performing parametric studies that require solutions for many values of normal loads.

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