scholarly journals Large elastic deformations of isotropic materials. III. Some simple problems in cyclindrical polar co-ordinates

1948 ◽  
Vol 240 (823) ◽  
pp. 509-525 ◽  
Keyword(s):  
Large Deformations ◽  
Equations Of Motion ◽  
Compressive Force ◽  
Cylindrical Tube ◽  
Cylindrical Symmetry ◽  
Simple Extension ◽  
Solid Cylinder ◽  
Normal Surface ◽  
Incompressibility Condition ◽  
Special Cases

Expressions for the components of strain and the incompressibility condition, for large deformations, are obtained in a cylindrical polar co-ordinate system. The stress-strain relations, equations of motion and boundary conditions for an incompressible, neo-Hookean material, in such a coordinate system, are also obtained and specialized to the case of cylindrical symmetry. These results are applied to the special cases of the simple torsion of a solid cylinder and of a hollow, cylindrical tube and to their combined simple extension and simple torsion. In the case of a solid cylinder, it is found that a state of simple torsion can be maintained by surface tractions applied to the ends of the cylinder only, and these consist of a torsional couple together with a compressive force. The necessary torsional couple is proportional to the amount of torsion and the compressive force to the square of the torsion. In the case of a hollow, cylindrical tube, it is again necessary to exert a torsional couple, proportional to the torsion, and a compressive force, proportional to the square of the torsion, on the plane ends, but it is also necessary to exert a normal surface traction, acting in a positive radial direction, on one or other of the curved surfaces of the tube and proportional to the square of the torsion.

scholarly journals Relativistic equations for elementary particles

1948 ◽  
Vol 192 (1029) ◽  
pp. 195-218 ◽  
Keyword(s):  
Elementary Particle ◽  
Wave Equation ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Wave Form ◽  
Total Charge ◽  
Field Equations ◽  
Relativistic Approximation ◽  
Special Cases ◽  
Linear Differential

From the general principles of quantum mechanics it is deduced that the wave equation of a particle can always be written as a linear differential equation of the first order with matrix coefficients. The principle of relativity and the elementary nature of the particle then impose certain restrictions on these coefficient matrices. A general theory for an elementary particle is set up under certain assumptions regarding these matrices. Besides, two physical assumptions concerning the particle are made, namely, (i) that it satisfies the usual second-order wave equation with a fixed value of the rest mass, and (ii) either the total charge or the total energy for the particle-field is positive definite. It is shown that in consequence of (ii) the theory can be quantized in the interaction free case. On introducing electromagnetic interaction it is found that the particle exhibits a pure magnetic moment in the non-relativistic approximation. The well-known equations for the electron and the meson are included as special cases in the present scheme. As a further illustration of the theory the coefficient matrices corresponding to a new elementary particle are constructed. This particle is shown to have states of spin both 3/2 and 1/2. In a certain sense it exhibits an inner structure in addition to the spin. In the non-relativistic approximation the behaviour of this particle in an electromagnetic field is the same as that of the Dirac electron. Finally, the transition from the particle to the wave form of the equations of motion is effected and the field equations are given in terms of tensors and spinors.

scholarly journals Large deformations of a cylindrical tube with prestressed coatings

2019 ◽  
Vol 484 (5) ◽  
pp. 547-549
Author(s):  
Yu. N. Kulchin ◽  
V. E. Ragozina ◽  
O. V. Dudko
Keyword(s):  
Shock Waves ◽  
Large Deformations ◽  
Cylindrical Tube ◽  
Plastic Strains ◽  
Rotation Tensor ◽  
Plastic Deformations ◽  
Incompressible Materials ◽  
Isotropic Media ◽  
Elastic Loading ◽  
Elastic Shock

General theoretical relations for calculating the redistribution of the preliminary irreversible strain field during unloading or elastic loading of a medium are obtained for the nonlinear multiplicative gradient model of large elastic-plastic deformations. It is shown that the dynamics of elastic shock waves does not depend directly on the previously accumulated plastic strains. A formula for the plastic-strain rotation tensor is obtained. It is shown that rigid rotation of plastic strains under elastic shock waves can be jump-like. All results are obtained for the general case of model relations of isotropic media and are valid for both compressible and incompressible materials.

Investigation of the Stability of Axially Moving Beams With Discrete Masses

2021 ◽  
Author(s):  
Konstantina Ntarladima ◽  
Michael Pieber ◽  
Johannes Gerstmayr
Keyword(s):  
Large Deformations ◽  
Equations Of Motion ◽  
Arbitrary Lagrangian Eulerian ◽  
Axial Motion ◽  
Multibody Modeling ◽  
Conveyor Belts ◽  
Axially Moving Beams ◽  
Concentrated Masses ◽  
Axially Moving ◽  
The Stability

Abstract The present paper addresses axially moving beams with co-moving concentrated masses while undergoing large deformations. For the numerical modeling, a novel beam finite element is introduced, which is based on the absolute nodal coordinate formulation extended with an additional Eulerian coordinate to represent the axial motion. The resulting formulation is well known as Arbitrary Lagrangian Eulerian (ALE) method, which is often used for axially moving beams and pipes conveying fluids. As compared to previous formulations, the present formulation allows us to introduce the Eulerian part by an independent coordinate, which fully incorporates the dynamics of the axial motion, while the shape functions remain independent of the beam coordinates and are thus constant. The proposed approach, which is derived from an extended version of Lagrange’s equations of motion, allows for the investigation of the stability of axially moving beams for a certain axial velocity and stationary state of large deformation. A multibody modeling approach allows us to extend the beam formulation for co-moving discrete masses, which represent concentrated masses attached to the beam, e.g., gondolas in ropeway systems, or transported masses in conveyor belts. Within numerical investigations we show that a larger number of discrete masses behaves similarly as the case of (continuously) distributed mass along the beam.

Dynamic Modeling of the Double-Nut Planetary Roller Screw Mechanism Considering Elastic Deformations

2021 ◽  
Author(s):  
Xiaojun Fu ◽  
Geng Liu ◽  
Xin Li ◽  
Ma Shangjun ◽  
Qiao Guan
Keyword(s):  
External Force ◽  
Load Distribution ◽  
Equations Of Motion ◽  
Great Influence ◽  
Compressive Force ◽  
Mass System ◽  
Elastic Deformations ◽  
Roller Screw ◽  
Screw Mechanism ◽  
Nonlinear Equations Of Motion

Abstract With the rising application of double-nut Planetary Roller Screw Mechanism (PRSM) into industry, increasing comprehensive studies are required to identify the interactions among motion, forces and deformations of the mechanism. A dynamic model of the double-nut PRSM with considering elastic deformations is proposed in this paper. As preloads, inertial forces and elastic deformations have a great influence on the load distribution among threads, the double-nut PRSM is discretized into a spring-mass system. An adjacency matrix is introduced, which relates the elastic displacements of nodes and the deformations of elements in the spring-mass system. Then, the compressive force acting on the spacer is derived and the equations of load distribution are given. Considering both the equilibrium of forces and the compatibility of deformations, nonlinear equations of motion for the double-nut PRSM are developed. The effectiveness of the proposed model is verified by comparing dynamic characteristics and the load distribution among threads with those from the previously published models. Then, the dynamic analysis of a double-nut PRSM is carried out, when the rotational speed of the screw and the external force acting on the nut #2 are changed periodically. The results show that if the external force is increased, the preload of the nut #1 is decreased and that of the nut #2 is increased. Although the nominal radii of rollers are the same, the maximum contact force acting on the roller #2 is much larger than that of the roller #1.

Modeling, Simulation, and Modal Analysis Hydraulic Valve Lifter With Oil Compressibility Effects

1989 ◽  
Author(s):  
T. Hatch ◽  
A. P. Pisano
Keyword(s):  
Differential Equations ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Check Valve ◽  
Linear Differential Equations ◽  
Third Order ◽  
Mode Behavior ◽  
Special Cases ◽  
Linear Differential ◽  
Hydraulic Valve

Abstract A two-degree-of-freedom (2-DOF), analytical model of a hydraulic valve lifter is derived. Special features of the model include the effects of bulk oil compressibility, multi-mode behavior due to plunger check valve modeling, and provision for the inclusion of third and fourth body displacements to aid In the use of the model in extended, multi-DOF systems. It is shown that motion of the lifter plunger and body must satisfy a coupled system of third-order, non-linear differential equations of motion. It is also shown that the special cases of zero oil compressibility and/or 1-DOF motion of lifter plunger can be obtained from the general third-order equations. For the case of zero oil compressibility, using Newtonian fluid assumptions, the equations of motion are shown to reduce to a system of second-order, linear differential equations. The differential equations are numerically integrated in five scenarios designed to test various aspects of the model. A modal analysis of the 2-DOF, compressible model with an external contact spring is performed and is shown to be in excellent agreement with simulation results.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

The Isobars in Boundary Layers at Supersonic Speeds

1968 ◽  
Vol 19 (2) ◽  
pp. 105-126 ◽  
Author(s):  
D. F. Myring ◽  
A. D. Young
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Equations Of Motion ◽  
Normal Pressure ◽  
Simple Extension ◽  
Pressure Gradients ◽  
Boundary Layer Flows ◽  
Hyperbolic Flow ◽  
The Individual ◽  
Momentum Integral

SummaryFor boundary layer flows over curved surfaces at moderately high supersonic speeds the existence of normal pressure gradients within the boundary layer becomes important even for small curvatures and they cannot be ignored. The describing equations are basically parabolic in form so that the simplifications inherent in hyperbolic flows would not at first sight seem to be relevant. However, the equations of motion for a two-dimensional, supersonic, rotational, viscous flow are analysed along the lines of a hyperbolic flow and the individual effects of viscosity and vorticity are examined with regard to the isobar distributions. It is found that these two properties have compensating effects and the experimental evidence presented confirms the conclusion that inside the boundary layer the isobars follow much the same rules as those which determine the isobars in the external hyperbolic flow. Since for turbulent boundary layers the fullness of the Mach number profile produces almost linear Mach lines in the boundary layer, this provides a simple extension to the methods of analysis, and the momentum integral equation is reformulated using a swept element bounded by linear isobars. The final equation is similar in form to the conventional one except that the momentum and displacement thicknesses are now defined by integrals along the swept isobars, and all normal pressure gradients due to centrifugal effects are accounted for.

Dynamics of the Elastica With End Mass and Follower Loading

1990 ◽  
Vol 57 (1) ◽  
pp. 203-208 ◽  
Author(s):  
J. M. Snyder ◽  
J. F. Wilson
Keyword(s):  
Internal Pressure ◽  
Large Deformations ◽  
Equations Of Motion ◽  
Dynamic Responses ◽  
Cantilever Beams ◽  
Static Responses ◽  
Tube Type ◽  
Payload Mass ◽  
Nonlinear Equations Of Motion ◽  
Special Case

Orthotropic, polymeric tubes subjected to internal pressure may undergo large deformations while maintaining linear moment-curvature behavior. Such tubes are modeled herein as inertialess, elastic cantilever beams (the elastica) with a payload mass at the tip and with internal pressure as the eccentric tip follower loading that drives the configurations through large deformations. From the nonlinear equations of motion, dynamic beam trajectories are calculated over a range of system parameters for the special case of a point mass at the tip and a terminated ramp pressure loading. The dynamic responses, which are unique because the loading history and the range of motion are fully defined, are presented in nondimensional form and are compared to static responses presented in a companion study. These results are applicable to the dynamic design of high flexure, tube-type, robotic manipulator arms.

Vibration Characteristics of Straight Blades of Asymmetrical Aerofoil Cross-Section

1969 ◽  
Vol 20 (2) ◽  
pp. 178-190 ◽  
Author(s):  
W. Carnegie ◽  
B. Dawson
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Limited Range ◽  
First Order ◽  
Special Cases ◽  
Theoretical Results ◽  
Theoretical Procedure ◽  
Good Agreement

SummaryTheoretical and experimental natural frequencies and modal shapes up to the fifth mode of vibration are given for a straight blade of asymmetrical aerofoil cross-section. The theoretical procedure consists essentially of transforming the differential equations of motion into a set of simultaneous first-order equations and solving them by a step-by-step finite difference procedure. The natural frequency values are compared with results obtained by an analytical solution and with standard solutions for certain special cases. Good agreement is shown to exist between the theoretical results for the various methods presented. The equations of motion are dependent upon the coordinates of the axis of the centre of flexure of the beam relative to the centroidal axis. The effect of variations of the centre of flexure coordinates upon the frequencies and modal shapes is shown for a limited range of coordinate values. Comparison is made between the theoretical natural frequencies and modal shapes and corresponding results obtained by experiment.

Compressibility Effects in High Speed Air Flow

1931 ◽  
Vol 35 (247) ◽  
pp. 665-674
Author(s):  
S. G. Hooker
Keyword(s):  
Angular Velocity ◽  
Perfect Fluid ◽  
Mathematical Analysis ◽  
High Speed ◽  
Equations Of Motion ◽  
Centre Of Gravity ◽  
High Speeds ◽  
Real Fluid ◽  
Special Cases ◽  
Compressibility Effects

Any theoretical attempt to evaluate the forces and couples experienced by an aircraft when in flight by a mathematical analysis of the pressures exerted by the air when in motion about the various parts, leads to what have, so far, proved insuperable difficulties. It involves the integration of the equations of motion of a real fluid, and, except in a few very special cases, these have been insoluble. The actual motion of a fluid is affected by a number of its properties, and, in general, accounts would have to be taken of its density, viscosity, and at high speeds its compressibility. In certain circumstances the effect of these last two can be neglected and the classical theory of hydrodynamics dealing with the motion of a non-viscous, incompressible or perfect fluid can be applied. A further simplification consists in supposing that the motion is irrotational, that is, any small portion of the fluid at a point has no angular velocity about its centre of gravity.

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