On the Elastic Bending of Columns Due to Dynamic Axial Forces Including Effects of Axial Inertia

1960 ◽  
Vol 27 (1) ◽  
pp. 125-131 ◽  
Author(s):  
Eugene Sevin
Keyword(s):  
Numerical Solutions ◽  
Equations Of Motion ◽  
Sine Wave ◽  
Elastic Response ◽  
Secondary Importance ◽  
Inertia Effects ◽  
Elastic Bending ◽  
Axial Forces ◽  
Compression Members ◽  
Constant Rate

This study is concerned with the influence of axial inertia upon the elastic bending motion of initially slightly curved columns acted on by time-dependent axial forces. The equations of motion include both axial inertia and nonlinear strain terms. Numerical solutions were obtained for a similar problem previously studied by Hoff [1] but in which axial-inertia effects were neglected; i.e., the problem of a simply supported column initially bent in the shape of a half sine wave and loaded by displacing one end axially at a constant rate. The range of solutions pertains to conventional structural compression members (slenderness ratios less than 150), and to minimum rates of loading compatible with elastic response of common engineering materials. This study suggests that axial-inertia effects are of secondary importance in so far as the gross elastic response of conventional structural columns is concerned.

High Speed Rail Short Bridge-Track-Train Interaction Based on the Decoupled Equations of Motion in the Finite Element Domain

2016 ◽  
Author(s):  
Said I. Nour ◽  
Mohsen A. Issa
Keyword(s):  
Finite Element ◽  
High Speed ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Rotational Inertia ◽  
Force Model ◽  
Inertia Effects ◽  
Contact Points ◽  
Short Span ◽  
Track Bridge

The interaction between the train, track, and bridge was considered as an interaction between two decoupled subsystems. A first subsystem consisted of the train vehicle simulated as a four-wheelset mass-spring-damper system having two layers of suspensions and ten degrees of freedom. A second subsystem consisted of the track-bridge system assumed to be a top rail beam and a bottom bridge beam coupled by continuous springs and dampers representing the elastic properties of the trackbed smeared over the spacing of the railway ties. The bridge supports were assumed to be rigid or flexible. The equations of motion of a finite element form were derived for each subsystem independently by means of the Newton’s second law. The dynamic interaction between the moving vehicle of the first subsystem and the stationary underlying track-bridge structure of the second subsystem was established by means of a no-separation constraint equation in the contact points between the wheels and the rails. The proposed two-dimensional analysis was intended to accurately describe the vertical behavior of short span bridges subjected to high-frequency excitations due to the passage of high speed trains; therefore, shear deformations, rotational inertia effects, and consistent mass matrices were adopted in the mathematical model. Numerical solutions of the decoupled equations of motion for both subsystems were obtained with the step-by-step direct integration in the time domain using HHT alpha method with a special scheme in the contact interface. The solution accuracy of the proposed method was validated against responses obtained from a semi-analytical method of a train car travelling over a simply supported bridge. The practical engineering application was demonstrated with a case study investigating effects of key parameters in the behavior of a ballasted short span railway bridge. Compared with the moving force model, results showed that for bridges with rigid supports both the vehicle interaction and trackbed produce lower peak responses at resonance speeds with the latter being more significant. However an increase in support flexibility had a greater impact across all speeds in increasing the bridge responses.

Dynamic Characteristics of a Hydrostatic Gas Bearing Driven by Oscillating Exhaust Pressure

1984 ◽  
Vol 106 (4) ◽  
pp. 477-483 ◽  
Author(s):  
C. B. Watkins ◽  
H. D. Branch ◽  
I. E. Eronini
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Source Model ◽  
Regular Perturbation ◽  
Response Characteristics ◽  
Finite Difference Approximations ◽  
Bode Plots ◽  
Gas Bearing

Vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating exhaust pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and sleeve is solved together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The numerical solutions of the perturbation equations are obtained by discretizing the pressure field using finite-difference approximations with a discrete, nonuniform line-source model which excludes effects due to feeding hole volume. An iterative scheme is used to simultaneously satisfy the equations of motion for the journal. The results presented include Bode plots of bearing-oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency.

scholarly journals Dynamics of nanodust particles emitted from elongated initial orbits

2018 ◽  
Vol 617 ◽  
pp. A43 ◽  
Author(s):  
A. Czechowski ◽  
I. Mann
Keyword(s):  
Solar Wind ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Dust Cloud ◽  
Theoretical Models ◽  
Ecliptic Plane ◽  
Parent Body ◽  
The Sun ◽  
Trapping Region ◽  
Eccentric Orbits

Context. Because of high charge-to-mass ratio, the nanodust dynamics near the Sun is determined by interplay between the gravity and the electromagnetic forces. Depending on the point where it was created, a nanodust particle can either be trapped in a non-Keplerian orbit, or escape away from the Sun, reaching large velocity. The main source of nanodust is collisional fragmentation of larger dust grains, moving in approximately circular orbits inside the circumsolar dust cloud. Nanodust can also be released from cometary bodies, with highly elongated orbits. Aims. We use numerical simulations and theoretical models to study the dynamics of nanodust particles released from the parent bodies moving in elongated orbits around the Sun. We attempt to find out whether these particles can contribute to the trapped nanodust population. Methods. We use two methods: the motion of nanodust is described either by numerical solutions of full equations of motion, or by a two-dimensional (heliocentric distance vs. radial velocity) model based on the guiding-center approximation. Three models of the solar wind are employed, with different velocity profiles. Poynting–Robertson and the ion drag are included. Results. We find that the nanodust emitted from highly eccentric orbits with large aphelium distance, like those of sungrazing comets, is unlikely to be trapped. Some nanodust particles emitted from the inbound branch of such orbits can approach the Sun to within much shorter distances than the perihelium of the parent body. Unless destroyed by sublimation or other processes, these particles ultimately escape away from the Sun. Nanodust from highly eccentric orbits can be trapped if the orbits are contained within the boundary of the trapping region (for orbits close to ecliptic plane, within ~0.16 AU from the Sun). Particles that avoid trapping escape to large distances, gaining velocities comparable to that of the solar wind.

scholarly journals Perturbation Solutions For Magnetohydrodynamics (Mhd) Flow of in a Non-Newtonian Fluid Between Concentric Cylinders

2019 ◽  
Vol 24 (1) ◽  
pp. 199-211
Author(s):  
M. Yürüsoy ◽  
Ö.F. Güler
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Variable Viscosity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Mhd Flow ◽  
Model Space ◽  
Viscosity Model ◽  
Concentric Cylinders ◽  
Approximate Analytical Solutions

Abstract The steady-state magnetohydrodynamics (MHD) flow of a third-grade fluid with a variable viscosity parameter between concentric cylinders (annular pipe) with heat transfer is examined. The temperature of annular pipes is assumed to be higher than the temperature of the fluid. Three types of viscosity models were used, i.e., the constant viscosity model, space dependent viscosity model and the Reynolds viscosity model which is dependent on temperature in an exponential manner. Approximate analytical solutions are presented by using the perturbation technique. The variation of velocity and temperature profile in the fluid is analytically calculated. In addition, equations of motion are solved numerically. The numerical solutions obtained are compared with analytical solutions. Thus, the validity intervals of the analytical solutions are determined.

Dynamic Rocking Response and Optimization of the Nonlinear Suspension of a Railroad Freight Car

1980 ◽  
Vol 102 (1) ◽  
pp. 86-93 ◽  
Author(s):  
M. Samaha ◽  
T. S. Sankar
Keyword(s):  
Equations Of Motion ◽  
Sine Wave ◽  
Optimization Techniques ◽  
Model Accuracy ◽  
Frequency Responses ◽  
Optimum Parameters ◽  
Freight Car ◽  
Six Degree Of Freedom ◽  
Railroad Freight ◽  
Rocking Response

A modified mathematical model of a large capacity railroad freight vehicle is presented. The model for this investigation is constructed in such a way to describe the bounce, sway and rocking modes of the system and also to account for most of the vehicle non-linearity effects. Equations of motion of the six degree of freedom nonlinear model are derived assuming that the excitations from the track in vertical and lateral directions are purely periodic in the form of a rectified sine wave. The solution for the time and frequency responses on digital computer are compared with available measured data to investigate the model accuracy. Multivariable optimization techniques are employed to find the optimum suspension parameters that minimizes the maximum car rocking response over the frequency range of interest. The optimum parameters are presented in different forms either for the existing or for stabilized vehicle configuration.

scholarly journals Bending–torsional-coupled vibrations and buckling characteristics of single and double composite Timoshenko beams

2019 ◽  
Vol 11 (3) ◽  
pp. 168781401983445
Author(s):  
Ma’en S Sari ◽  
Wael G Al-Kouz ◽  
Rafat Al-Waked
Keyword(s):  
Slenderness Ratio ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
Engineering Structures ◽  
Axial Forces ◽  
Double Beam ◽  
Tensile Forces ◽  
Torsional Coupling ◽  
The Stability

The stability and free vibration analyses of single and double composite Timoshenko beams have been investigated. The closed-section beams are subjected to constant axially compressive or tensile forces. The double beams are assumed to be connected by a layer of elastic translational and rotational springs. The coupled governing partial differential equations of motion are discretized, and the resulted eigenvalue problem is solved numerically by applying the Chebyshev spectral collocation method. The effects of the elastic layer parameters, the axial forces, the slenderness ratio, the bending–torsional coupling, and the boundary conditions on the critical buckling loads, mode shapes, and natural transverse frequencies have been studied. A parametric study was performed, and the obtained results revealed different features, which hopefully can be useful for single- and double-beam-like engineering structures.

scholarly journals Resonant Stellar Orbits in Spiral Galaxies

1975 ◽  
Vol 69 ◽  
pp. 237-244
Author(s):  
P. O. Vandervoort
Keyword(s):  
Excellent Agreement ◽  
Spiral Galaxy ◽  
Harmonic Balance ◽  
Spiral Galaxies ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
Stellar Orbits ◽  
Method Of Harmonic Balance ◽  
Resonance Phenomena

This paper reviews a series of investigations of the orbits of stars in the regions of the Lindblad resonances of a spiral galaxy. The analysis is formulated in an epicyclic approximation. Analytic solutions of the epicyclic equations of motion are obtained by the method of harmonic balance of Bogoliubov and Mitropolsky. These solutions represent the resonance phenomena exhibited by the orbits in generally excellent agreement with numerical solutions.

scholarly journals The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
T. S. Amer
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Dynamical Behavior ◽  
Vertical Plane ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Model

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

Nonlinear Dynamics of Rotating Structures and Helicopter Blades

2018 ◽  
Author(s):  
Matteo Filippi ◽  
Alfonso Pagani ◽  
Erasmo Carrera
Keyword(s):  
Numerical Solutions ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Constitutive Law ◽  
Rotating Blades ◽  
Helicopter Blades ◽  
Strain Components ◽  
Theory Approximation ◽  
Nonlinear Equations Of Motion ◽  
Lagrange Strain

This work explores the effects of geometrical nonlinearities in the vibration analysis of rotating structures and helicopter blades. Structures are modelled via higher-order beam theories with variable kinematics. These theories fall in the domain of the Carrera Unified Formulation (CUF), according to which the nonlinear equations of motion of rotating blades can be written in terms of fundamental nuclei, whose formalism is an invariant of the theory approximation. The inherent three-dimensional nature of CUF enables one to include all Green-Lagrange strain components as well as all coupling effects due to the geometrical features and the three-dimensional constitutive law. Numerical solutions are considered and opportunely discussed. Also, linearized and full nonlinear solutions for vibrating rotating blades are compared both in case of small amplitudes and in the large deflections/rotations regime.

Structure and Propagation of Rotating Stall in a Single- and a Multi-Stage Axial Compressor

1999 ◽  
Author(s):  
H. M. Saxer-Felici ◽  
A. P. Saxer ◽  
F. Ginter ◽  
A. Inderbitzin ◽  
G. Gyarmathy
Keyword(s):  
Numerical Solutions ◽  
Equations Of Motion ◽  
Axial Compressor ◽  
Rotating Stall ◽  
Water Model ◽  
Two Stage ◽  
Additional Information ◽  
Driving Mechanisms ◽  
Multi Stage ◽  
Stall Cells

The structure and propagation of rotating stall cells in a single- and a two-stage subsonic axial compressor is addressed in this paper using computational and experimental analysis. Unsteady solutions of the 2-D inviscid compressible (Euler) equations of motion are presented for one operating point in the fully-developed rotating stall regime for both a single- and a two-stage compressor. The inviscid assumption is verified by comparing the single-stage 2-D in viscid/compressible solution with an equivalent 2-D viscous (Navier-Stokes) result for incompressible flow. The structure of the rotating stall cell is analyzed and compared for the single- and two-stage cases. The numerical solutions are validated against experimental data consisting of flow visualization and unsteady row-by-row static pressure measurements obtained in a four-stage water model of a subsonic compressor. The CFD solutions supply a link between the observed experimental features and provide additional information on the structure of the stall flow. Based on this study. supporting assumptions regarding the driving mechanisms for the propagation of fully-developed rotating stall cells and their structure are postulated. In methodical respect the results suggest that the inviscid model is able to reproduce the essentials of the flow physics associated with the propagation of fully-developed, full-span rotating stall in a subsonic axial compressor.

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