Energy Conserving Equations of Motion for Gear Systems

2004 ◽  
Vol 127 (2) ◽  
pp. 208-212 ◽  
Author(s):  
Sejoong Oh ◽  
Karl Grosh ◽  
James R. Barber
Keyword(s):  
Equations Of Motion ◽  
Interaction Force ◽  
Fundamental Principle ◽  
Step Change ◽  
Conservation Of Energy ◽  
Gear Design ◽  
Contact Points ◽  
Energy Conserving ◽  
External Forces ◽  
Work Done

A system of two meshing gears exhibits a stiffness that varies with the number of teeth in instantaneous contact and the location of the corresponding contact points. A classical Newtonian statement of the equations of motion leads to a solution that contradicts the fundamental principle of mechanics that the change in total energy in the system is equal to the work done by the external forces, unless the deformation of the teeth is taken into account in defining the direction of the instantaneous tooth interaction force. This paradox is avoided by using a Lagrange’s equations to derive the equations of motion, thus ensuring conservation of energy. This introduces nonlinear terms that are absent in the classical equations of motion. In particular, the step change in stiffness associated with the introduction of an additional tooth to contact implies a step change in strain energy and hence a corresponding step change in kinetic energy and rotational speed. The effect of these additional terms is examined by dynamic simulation, using a system of two involute spur gears as an example. It is shown that the two systems of equations give similar predictions at high rotational speeds, but they differ considerably at lower speeds. The results have implications for gear design, particularly for low speed gear sets.

Dynamic Behavior of Damaged Bridge with Multi-Cracks Under Moving Vehicular Loads

2017 ◽  
Vol 17 (02) ◽  
pp. 1750019 ◽  
Author(s):  
Xinfeng Yin ◽  
Yang Liu ◽  
Lu Deng ◽  
Xuan Kong
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Coupled System ◽  
Interaction Force ◽  
Vehicle Model ◽  
Coupled Equations ◽  
Contact Points ◽  
Local Flexibility ◽  
Bridge Structures ◽  
Road Surface Roughness

When studying the vibration of a bridge–vehicle coupled system, most researchers mainly focus on the intact or original bridge structures. Nonetheless, a large number of bridges were built long ago, and most of them have suffered serious deterioration or damage due to the increasing traffic loads, environmental effect, material aging, and inadequate maintenance. Therefore, the effect of damage of bridges, such as cracks, on the vibration of vehicle–bridge coupled system should be studied. The objective of this study is to develop a new method for considering the effect of cracks and road surface roughness on the bridge response. Two vehicle models were introduced: a single-degree-of-freedom (SDOF) vehicle model and a full-scale vehicle model with seven degrees of freedom (DOFs). Three typical bridges were investigated herein, namely, a single-span uniform beam, a three-span stepped beam, and a non-uniform three-span continuous bridge. The massless rotational spring was adopted to describe the local flexibility induced by a crack on the bridge. The coupled equations for the bridge and vehicle were established by combining the equations of motion for both the bridge and vehicles using the displacement relationship and interaction force relationship at the contact points. The numerical results show that the proposed method can rationally simulate the vibrations of the bridge with cracks under moving vehicular loads.

scholarly journals Energy Considerations in Systems With Varying Stiffness

2003 ◽  
Vol 70 (4) ◽  
pp. 465-469 ◽  
Author(s):  
J. R. Barber ◽  
K. Grosh ◽  
S. Oh
Keyword(s):  
Energy Conservation ◽  
Total Energy ◽  
Dynamic Behavior ◽  
Contact Force ◽  
Elastic System ◽  
Equations Of Motion ◽  
Local Deformation ◽  
Elastic Systems ◽  
External Forces ◽  
Work Done

If the stiffness of an elastic system changes with time, a conventional Newtonian statement of the equations of motion will generally lead to solutions that violate the fundamental mechanics principle that the work done by the external forces be equal to the increase in total energy of the system. Timoshenko’s discussion of the problem of a vehicle driven across an elastic bridge is generalized to show that energy conservation can be restored only if the local deformation of the components is taken into account in determining the direction of the contact force. This result has important consequences for the interaction of elastic systems in general, including, for example, the dynamic behavior of meshing gears.

Microstructure Theory for a Composite Beam

1971 ◽  
Vol 38 (4) ◽  
pp. 947-954 ◽  
Author(s):  
C.-T. Sun
Keyword(s):  
Composite Beam ◽  
Continuum Model ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Present Theory ◽  
Flexural Waves ◽  
Subsequent Application ◽  
External Forces ◽  
Work Done ◽  
Constituent Layer

A continuum model with microstructure is constructed for laminated beams. In deriving equations, each constituent layer is considered as a Timoshenko beam. With a certain kinematical assumption regarding the deformations of the composite beam the kinetic and strain energies, as well as the variation of the work done by external forces, are computed. The energy and work expressions are “smoothed out” by a smoothing operation, thus transforming the laminated structuring into microstructure of a macro-homogeneous beam. Subsequent application of Hamilton’s principle yields the equations of motion and the boundary conditions. The equations thus obtained are employed to investigate free flexural waves in a composite beam. It is found that the dispersion curves according to the present theory agree very well with the curves obtained according to an exact analysis. Results according to the effective modulus theory are presented for comparison. Two simplified versions of the microstructure beam theory are also developed and discussed.

scholarly journals OBTAINING THE EQUATIONS OF MOTION OF A MECHANICAL BODY THROUGH ANALYTICAL METHODS

2021 ◽  
Author(s):  
Armandt Erasmus
Keyword(s):  
Dynamical System ◽  
Analytical Methods ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Lagrangian Mechanics ◽  
Conservation Of Energy ◽  
Least Action ◽  
Principle Of Least Action ◽  
Action Value ◽  
External Forces

The aim of this paper is to obtain the equations of motion in n-dimensional space for the case where no external forces act on a mechanical system using analytical methods. One such method is known as Lagrangian Mechanics. Lagrangian Mechanics is founded on the principle of least action which states that the spontaneous change from one configuration to another of a dynamical system has a minimum action value if the law of conservation of energy holds.

Redox Reactions in Lipid Membranes as a Model for Primordial Energy-Conserving Systems

1997 ◽  
Vol 161 ◽  
pp. 437-442
Author(s):  
Salvatore Di Bernardo ◽  
Romana Fato ◽  
Giorgio Lenaz
Keyword(s):  
Experimental Evidence ◽  
Lipid Membranes ◽  
Energy Supply ◽  
Redox Reactions ◽  
Living Systems ◽  
Asymmetric Distribution ◽  
Conservation Of Energy ◽  
Asymmetrical Distribution ◽  
Energy Conserving ◽  
Living Organisms

AbstractOne of the peculiar aspects of living systems is the production and conservation of energy. This aspect is provided by specialized organelles, such as the mitochondria and chloroplasts, in developed living organisms. In primordial systems lacking specialized enzymatic complexes the energy supply was probably bound to the generation and maintenance of an asymmetric distribution of charged molecules in compartmentalized systems. On the basis of experimental evidence, we suggest that lipophilic quinones were involved in the generation of this asymmetrical distribution of charges through vectorial redox reactions across lipid membranes.

scholarly journals Numerical methods for General Relativistic particles

2018 ◽  
Vol 14 (S342) ◽  
pp. 19-23
Author(s):  
Fabio Bacchini ◽  
Bart Ripperda ◽  
Alexander Y. Chen ◽  
Lorenzo Sironi
Keyword(s):  
Numerical Methods ◽  
Gravitational Lensing ◽  
Complex Dynamics ◽  
Equations Of Motion ◽  
Numerical Algorithms ◽  
Relativistic Effects ◽  
Compact Objects ◽  
Light Rays ◽  
Massive Particles ◽  
External Forces

AbstractWe present recent developments on numerical algorithms for computing photon and particle trajectories in the surrounding of compact objects. Strong gravity around neutron stars or black holes causes relativistic effects on the motion of massive particles and distorts light rays due to gravitational lensing. Efficient numerical methods are required for solving the equations of motion and compute i) the black hole shadow obtained by tracing light rays from the object to a distant observer, and ii) obtain information on the dynamics of the plasma at the microscopic scale. Here, we present generalized algorithms capable of simulating ensembles of photons or massive particles in any spacetime, with the option of including external forces. The coupling of these tools with GRMHD simulations is the key point for obtaining insight on the complex dynamics of accretion disks and jets and for comparing simulations with upcoming observational results from the Event Horizon Telescope.

Buckling of Multiple-Bay Ring-Reinforced Cylindrical Shells Subject to Hydrostatic Pressure

1953 ◽  
Vol 20 (4) ◽  
pp. 469-474
Author(s):  
W. A. Nash
Keyword(s):  
Cylindrical Shell ◽  
Hydrostatic Pressure ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Middle Surface ◽  
Elastic Buckling ◽  
Elastic Instability ◽  
Total Potential ◽  
External Forces ◽  
Work Done

Abstract An analytical solution is presented for the problem of the elastic instability of a multiple-bay ring-reinforced cylindrical shell subject to hydrostatic pressure applied in both the radial and axial directions. The method used is that of minimization of the total potential. Expressions for the elastic strain energy in the shell and also in the rings are written in terms of displacement components of a point in the middle surface of the shell. Expressions for the work done by the external forces acting on the cylinder likewise are written in terms of these displacement components. A displacement configuration for the buckled shell is introduced which is in agreement with experimental evidence, in contrast to the arbitrary patterns assumed by previous investigators. The total potential is expressed in terms of these displacement components and is then minimized. As a result of this minimization a set of linear homogeneous equations is obtained. In order that a nontrivial solution to this system of equations exists, it is necessary that the determinant of the coefficients vanish. This condition determines the critical pressure at which elastic buckling of the cylindrical shell will occur.

Unraveling Paradoxical Theories for Rigid Body Collisions

1991 ◽  
Vol 58 (4) ◽  
pp. 1049-1055 ◽  
Author(s):  
W. J. Stronge
Keyword(s):  
Frictional Force ◽  
Contact Point ◽  
Tangential Velocity ◽  
Velocity Slip ◽  
Rigid Bodies ◽  
Contact Points ◽  
Before And After ◽  
Limited Applicability ◽  
Work Done ◽  
Impulsive Reaction

A collision between two rigid bodies has a normal impulsive reaction at the contact point (CP). If the bodies are slightly rough and the contact points have a relative tangential velocity (slip), there is also a frictional force that opposes slip. Small initial slip can halt before contact terminates; when slip halts the frictional force changes and the collision process is separated into periods before and after halting. An energetically consistent theory for collisions with slip that halts is based on the work done by normal (nonfrictional) forces during restitution and compression phases. This theory clearly separates dissipation due to frictional forces from that due to internal irreversible deformation. With this theory, both normal and tangential components of the impulsive reaction always dissipate energy during collisions. In contrast, Newton’s impact law results in calculations of paradoxical increases in energy for collisions where slip reverses. This law relates normal components of relative velocity for the CP at separation and incidence by a constant (the coefficient of restitution e). Newton’s impact law is a kinematic definition for e that generally depends on the slip process and friction; consequently it has limited applicability.

Added Fluid Mass and the Equations of Motion of a Parachute

1983 ◽  
Vol 34 (3) ◽  
pp. 226-242 ◽  
Author(s):  
John A. Eaton
Keyword(s):  
Equations Of Motion ◽  
Added Mass ◽  
Unsteady Forces ◽  
Fluid Mass ◽  
Mass Tensor ◽  
Body Shapes ◽  
The Stability ◽  
External Forces ◽  
Six Degree Of Freedom ◽  
Nonlinear Solutions

SummaryWhile it has long been known that added fluid mass may be important in the dynamics of parachutes, due to inadequate or incorrect derivation and/or implementation of the added mass tensor its full significance in the stability of parachutes has yet to be appreciated. The concept of added mass is outlined and some general conditions for its significance are presented. Its implementation in the parachute equations of motion is reviewed, and the equations used in previous treatments are shown to be erroneous. A general method for finding the equivalent external forces and moments due to added mass is given, and the correct, anisotropic forms of the added mass tensor are derived for the six degree-of-freedom motion in an ideal fluid of rigid body shapes with planar-, twofold- and axisymmetry, These derivations may also be useful in dynamic stability studies of other low relative density bodies such as airships, balloons, submarines and torpedoes. Full nonlinear solutions of the equations of motion for the axisymmetric parachute have been obtained, and results indicate that added mass effects are more significant than previously predicted. In particular, the component of added mass along the axis of symmetry has a strong influence on stability. Better data on unsteady forces and moments on parachutes are needed.

Perturbations of Thermodynamic State Functions

2019 ◽  
pp. 124-131
Author(s):  
Robert H. Swendsen
Keyword(s):  
Statistical Mechanics ◽  
Thermodynamic Quantities ◽  
Conservation Of Energy ◽  
Important Concept ◽  
Thermodynamic State ◽  
Integrating Factor ◽  
Key Concepts ◽  
Small Change ◽  
Physical Phenomena ◽  
Work Done

Because small changes in thermodynamic quantities will play a central role in much of the development of thermodynamics, the key concepts are introduced in this short chapter. The First Law (conservation of energy) can be expressed simply in terms of infinitesimal quantities: a small change in the energy of a system is equal to the heat added plus the work done on the system. The theories of statistical mechanics and thermodynamics deal with the same physical phenomena. Exact and inexact differentials are defined, along with the important concept of an integrating factor that relates them. The useful equation relating small changes in heat to corresponding changes in entropy is derived.

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