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.

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.

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.

The influence of engine gyroscopic moments on vehicle transient handling

2021 ◽  
pp. 095440702199689
Author(s):  
Ali Shahabi ◽  
Amir Hossein Kazemian ◽  
Saeid Farahat ◽  
Faramarz Sarhaddi
Keyword(s):  
Coordinate System ◽  
Dynamic Behavior ◽  
Opposite Direction ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Vehicle Handling ◽  
Torque Vector ◽  
Crankshaft Rotation ◽  
External Forces ◽  
Engine Crankshaft

This study presents the dynamics of a 15-DOF model of the vehicle by performing simulations to investigate the vehicle handling dynamics in J-turn maneuver. Using Newton’s equations of motion, the equations of motion for the sprung and unsprung masses are all written in the vehicle coordinate system and the tire is modeled with the Pacejka 89 model. Since the engine crankshaft has a rotation relative to the vehicle coordinate system, in order to investigate the effect of the engine gyroscopic moments on the vehicle handling dynamics, the effect of the crankshaft rotation on the torque vector of external forces is considered. The direction of crankshaft rotation can change the direction of engine rotation in the direction of the wheels rotation or in the opposite direction of their rotation, which causes some changes in the gyroscopic torque vector of the engine. Due to the rotation speed of the crankshaft and its moment of inertia, the gyroscopic moments which resulted from the angular momentum of the engine crankshaft are considerable. These gyroscopic moments are added to the torque equation of external forces in the vehicle coordinate system and affect the vehicle handling dynamics. By using the numerical method of Newmark, vehicle’s dynamic behavior is investigated and the validation of its dynamic behavior is done by ADAMS/Car software. This study shows that in transverse engine, if the direction of engine rotation is in the opposite direction of the wheels, the vehicle handling dynamics is improved.

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.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

Hertz Contact Force Model With Permanent Indentation in Impact Analysis of Solids

1992 ◽  
Author(s):  
Hamid M. Lankarani ◽  
Parviz E. Nikravesh
Keyword(s):  
Contact Force ◽  
Impact Analysis ◽  
Equations Of Motion ◽  
Solid Particles ◽  
Hertzian Contact ◽  
Contact Period ◽  
Force Model ◽  
Unknown Parameters ◽  
Contact Force Model ◽  
Energy And Momentum

Abstract A continuous analysis method for the direct-central impact of two solid particles is presented. Based on the assumption that local plasticity effects are the sole factor accounting for the dissipation of energy in impact, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are analytically evaluated in terms of a given coefficient of restitution and velocities before impact. The equations of motion of the two solids may then be integrated forward in time knowing the variation of the contact force during the contact period. For Illustration, an impact of two soft metallic particles is studied.

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.

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