DISPLACEMENT-IMPULSE COMPLEMENTARY IN ANALYTICAL MECHANICS

1994 ◽  
Vol 18 (3) ◽  
pp. 225-247
Author(s):  
F.P.J. Rimrott ◽  
B. Tabarrok ◽  
J. Altenbach
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Mechanical System ◽  
Equations Of Motion ◽  
Linear Momentum ◽  
Analytical Mechanics ◽  
Translatory Motion ◽  
Generalized Displacements ◽  
Energy Aspects

Based on Euler’s linear momentum law, equations of motion of a mechanical system, consisting of a collection of point masses and force elements, such as springs or dashpots, are derived using generalized impulses, rather than generalized displacements, as coordinates. There are limitations to this approach involving both potential energy aspects and kinetic energy aspects. The present paper if of introductory nature and restricted to systems with translatory motion.

CENTRIFUGAL IMPULSE AS COORDINATE IN THE TABARROKIAN FORMULATION

1995 ◽  
Vol 19 (3) ◽  
pp. 261-269
Author(s):  
F.P.J. Rimrott ◽  
W.M. Szczygielski
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Centrifugal Force ◽  
Nonholonomic Constraint ◽  
Lagrange Equation ◽  
Equations Of Motion ◽  
Essential Elements ◽  
Model System ◽  
The Other ◽  
Generalized Displacements

While the well-known conventional Lagrange equation, based on kinetic coenergy and potential energy, uses generalized displacements of the inertia (mass) elements of a system as coordinates, the complementary alternative or Tabarrok formulation, is based on kinetic energy and potential coenergy, and uses as coordinates the generalized impulses of the system’s force (spring) elements. A model system specifically selected to be as simple as possible, yet to contain all essential elements for an illustration of the application of the Tabarrokian approach for the case where a centrifugal force is present, has been devised to show that the centrifugal impulse appears as additional coordinate for the complementary Lagrangian, and that the system turns out to be non-Tabarrokian. It is then shown that the centrifugal impulse is related to the other impulse coordinates by a nonholonomic constraint. Eventually the compatibility equations of motion for the model system are obtained.

scholarly journals A method of constructing programmed motions of a mechanical system

1998 ◽  
Vol 20 (3) ◽  
pp. 46-57
Author(s):  
Do Sanh
Keyword(s):  
Mechanical System ◽  
Equations Of Motion ◽  
Analytical Mechanics

In the paper the method for constructing programmed motion is represented. The requirements for the programmed motion are treated as ideal constraints in analytical mechanics. The programmed motions expressed in Lagrange coordinates and in quasi coordinates are investigated. By applying the form of equations of motion of a constrained mechanical system [7], a schema for calculating programmed motions has been established. By this schema the errors of realizing programmed motions have been reduced and controlled. For illustration of the method some examples have been investigated.

scholarly journals DIRECT HAMILTONIZATION — GENERALIZATION OF AN ALTERNATIVE HAMILTONIZATION

2012 ◽  
Vol 22 (06) ◽  
pp. 1250135
Author(s):  
MARIA LEWTCHUK ESPINDOLA
Keyword(s):  
Dynamical Systems ◽  
Mechanical System ◽  
Equations Of Motion ◽  
A Priori ◽  
Mechanical Systems ◽  
Hamiltonian Function ◽  
Analytical Mechanics ◽  
Principal Change

A new procedure named direct Hamiltonization presents an alternative foundation to Analytical Mechanics, since in this formalism the Hamiltonian function can be obtained for all mechanical systems. The principal change proposed in this procedure is that the conjugate momenta cannot be defined a priori, but are established as a consequence of a canonical description of the mechanical system. The direct Hamiltonization is a generalization of the alternative one, where the usual Hamiltonization and momenta are recovered whenever they exist. Also this procedure assures the existence of a Hamiltonian function without any constraints for any mechanical system, therefore the usual quantization is always allowed. This procedure can be applied to non-Lagrangian, Nambu, nonholonomic and dynamical systems since there are no restrictions in this formalism as, for example, the number of equations of motion.

scholarly journals Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

2021 ◽  
Author(s):  
Javier Bonet ◽  
Antonio J. Gil
Keyword(s):  
Kinetic Energy ◽  
Crack Propagation ◽  
Mathematical Models ◽  
Linear Elasticity ◽  
Strain Energy ◽  
Equations Of Motion ◽  
Two Dimensions ◽  
Discontinuous Solutions ◽  
Linear Elasticity Theory ◽  
Intersonic Crack Propagation

AbstractThis paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

Vibration analysis of multiple degrees of freedom mechanical system with dry friction

2012 ◽  
Vol 227 (7) ◽  
pp. 1505-1514 ◽  
Author(s):  
SD Yu ◽  
BC Wen
Keyword(s):  
Mechanical System ◽  
Dry Friction ◽  
Degrees Of Freedom ◽  
Simple Procedure ◽  
Mathematical Problem ◽  
Equations Of Motion ◽  
Inequality Constraints ◽  
Integration Scheme ◽  
Time Step ◽  
Multiple Degrees Of Freedom

This article presents a simple procedure for predicting time-domain vibrational behaviors of a multiple degrees of freedom mechanical system with dry friction. The system equations of motion are discretized by means of the implicit Bozzak–Newmark integration scheme. At each time step, the discontinuous frictional force problem involving both the equality and inequality constraints is successfully reduced to a quadratic mathematical problem or the linear complementary problem with the introduction of non-negative and complementary variable pairs (supremum velocities and slack forces). The so-obtained complementary equations in the complementary pairs can be solved efficiently using the Lemke algorithm. Results for several single degree of freedom and multiple degrees of freedom problems with one-dimensional frictional constraints and the classical Coulomb frictional model are obtained using the proposed procedure and compared with those obtained using other approaches. The proposed procedure is found to be accurate, efficient, and robust in solving non-smooth vibration problems of multiple degrees of freedom systems with dry friction. The proposed procedure can also be applied to systems with two-dimensional frictional constraints and more sophisticated frictional models.

Diffusion and dissipation by linear momentum in spherical environment

2014 ◽  
Vol 28 (11) ◽  
pp. 1450077
Author(s):  
Werner Scheid ◽  
Aurelian Isar ◽  
Aurel Sandulescu
Keyword(s):  
Potential Energy ◽  
Density Matrix ◽  
Finite Number ◽  
Bessel Functions ◽  
Open Quantum System ◽  
Equation Of Motion ◽  
Linear Momentum ◽  
Spherical Space ◽  
Infinite Energy ◽  
Infinite Wall

An open quantum system is studied consisting of a particle moving in a spherical space with an infinite wall. With the theory of Lindblad the system is described by a density matrix which gets affected by operators with diffusive and dissipative properties depending on the linear momentum and density matrix only. It is shown that an infinite number of basis states leads to an infinite energy because of the infinite unsteadiness of the potential energy at the infinite wall. Therefore only a solution with a finite number of basis states can be performed. A slight approximation is introduced into the equation of motion in order that the trace of the density matrix remains constant in time. The equation of motion is solved by the method of searching eigenvalues. As a side-product two sums over the zeros of spherical Bessel functions are found.

MULTI–INERT OSCILLATORY MECHANISM

2020 ◽  
Vol 2 ◽  
pp. 19-25
Author(s):  
I.P. POPOV
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Oscillation Frequency ◽  
Free Oscillation ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Harmonic Oscillations ◽  
Oscillatory Systems ◽  
Oscillatory Mechanism ◽  
Mutual Conversion

A mechanical oscillatory system with homogeneous elements, namely, with n massive loads (multi– inert oscillator), is considered. The possibility of the appearance of free harmonic oscillations of loads in such a system is shown. Unlike the classical spring pendulum, the oscillations of which are due to the mutual conversion of the kinetic energy of the load into the potential energy of the spring, in a multi–inert oscillator, the oscillations are due to the mutual conversion of only the kinetic energies of the goods. In this case, the acceleration of some loads occurs due to the braking of others. A feature of the multi–inert oscillator is that its free oscillation frequency is not fixed and is determined mainly by the initial conditions. This feature can be very useful for technical applications, for example, for self–neutralization of mechanical reactive (inertial) power in oscillatory systems.

Investigation of Parametric Resonance Instability in a Flexible Rod Slider Crank Mechanism

1991 ◽  
Author(s):  
David G. Beale ◽  
Shyr-Wen Lee
Keyword(s):  
Potential Energy ◽  
Parametric Resonance ◽  
Variational Approach ◽  
Equations Of Motion ◽  
Monodromy Matrix ◽  
Bending Energy ◽  
Constrained Equations ◽  
Beam Bending ◽  
Floating Frame ◽  
Crank Mechanism

Abstract A direct variational approach with a floating frame is presented to derive the ordinary differential equations of motion of a flexible rod, constant crank speed slider crank mechanism. Potential energy terms contained in the derivation include beam bending energy and energy in foreshortening of the rod tip (which were selected because of the importance of these terms in a pinned-pinned rod parametric resonance). A symbolic manipulator code is used to reduce the constrained equations of motion to unconstrained nonlinear equations. A linearized version of these equations is used to explore parametric resonance stability-instability zones at low crank speeds and small deflections by a monodromy matrix technique.

Tracker-assisted modelling: simultaneous validation of the conservation law of energy and the work-energy theorem

2021 ◽  
Vol 57 (1) ◽  
pp. 015012
Author(s):  
Unofre B Pili ◽  
Renante R Violanda
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Gravitational Potential ◽  
Conservation Law ◽  
Gravitational Potential Energy ◽  
Time Data ◽  
Home Based ◽  
Basic Physics ◽  
Free Falling ◽  
Work Done

Abstract The video of a free-falling object was analysed in Tracker in order to extract the position and time data. On the basis of these data, the velocity, gravitational potential energy, kinetic energy, and the work done by gravity were obtained. These led to a rather simultaneous validation of the conservation law of energy and the work–energy theorem. The superimposed plots of the kinetic energy, gravitational potential energy, and the total energy as respective functions of time and position demonstrate energy conservation quite well. The same results were observed from the plots of the potential energy against the kinetic energy. On the other hand, the work–energy theorem has emerged from the plot of the total work-done against the change in kinetic energy. Because of the accessibility of the setup, the current work is seen as suitable for a home-based activity, during these times of the pandemic in particular in which online learning has remained to be the format in some countries. With the guidance of a teacher, online or face-to-face, students in their junior or senior high school—as well as for those who are enrolled in basic physics in college—will be able to benefit from this work.

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