The motion of a point mass along a string

An equation for the trajectory of a point mass (a particle) when it moves (slides) in a plane by inertia along a weightless elastic thread (string), stretched between two fixed points is obtained. The time dependence of the trajectory parameter is established. An equation of the trajectory of the particle when it suddenly decelerates is obtained. Forced motion of the particle along a straight string (as a model of the swinging of a lift on an elastic tether in zero gravity) is considered.     

Cross-country skiing motion equations,locomotive forces and mass scaling laws

This article presents differential equations for locomotive force and velocity during cross-country skiing. A muscle's work power is modelled. Thereafter, a locomotive force that depends on the skier's velocity is constructed. The external forces aerodynamic drag, friction forces and the force of gravity are incorporated in order to provide the equation of motion. Some allometric mass scaling relations are established and used to analyse the effect of a skier's mass on velocity. The model is tested by using a GPS instrument. We compare analytically and experimentally determined skiing distances and velocities as functions of time, and under different conditions. The article provides tools useful for practising athletes and coaches.     

The motion of a point mass along a string

As a supplement to results obtained earlier [1], the general integral of motion of a point mass along a string is determined, and the influence of friction is evaluated.     

Straight chains of interconnected mass points under the action of non-symmetric dry friction

The motion of a chain of interconnected equal material points placed on a rough straight line and connected by equal kinematical constraints is considered. It is supposed that the system experiences a small non-symmetric Coulomb dry frictional force acting from the line side upon each mass point and opposite to the direction of motion. The magnitude of this force depends on the direction of motion. To the equation of motion the procedure of averaging is applied. The expression for the stationary “on the average” velocity of the motion of the system as a single whole is found. The obtained theoretical results can be used for the design of worm-like vibration robots. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximate formula for the vertical asymptote of projectile motion in midair

The classic problem of the motion of a point mass (projectile) thrown at an angle to the horizon is reviewed. The air drag force is taken into account with the drag factor assumed to be constant. An analytical approach is used for the investigation. An approximate formula is obtained for one of the characteristics of the motion – the vertical asymptote. The value of an asymptote is determined directly by the initial conditions of throwing. Analytically derived values of asymptotes in comparison with numerical values obtained by integrating the equations of motion are given. The motion of a baseball is presented as an example.     

The motion in a perfect fluid of a body containing a moving point mass

The motion of a system (a rigid body, symmetrical about three mutually perpendicular planes, plus a point mass situated inside the body) in an unbounded volume of a perfect fluid, which executes vortex-free motion and is at rest at infinity, is considered. The motion of the body occurs due to displacement of the point mass with respect to the body. Two cases are investigated: (a) there are no external forces, and (b) the system moves in a uniform gravity field. An analytical investigation of the dynamic equations under conditions when the point performs a specified plane periodic motion inside the body showed that in case (a) the system can be displaced as far as desired from the initial position. In case (b) it is proved that, due to the permanent addition of energy of the corresponding relative motion of the point, the body may float upwards. On the other hand, if the velocity of relative motion of the point is limited, the body will sink. The results of numerical calculations, when the point mass performs random walks along the sides of a plane square grid rigidly connected with the body, are presented.     

Optimization of the motion in a resistant medium of a body with a movable internal mass

The motion, in a resistant medium, of a system consisting of a rigid body and movable internal mass is considered. The external medium acts on the body by a force that piecewise linearly depends on its speed. The class of periodic motions of the internal mass for which the speed of this mass relative to the body is piecewise constant is studied. It is shown that, under certain conditions, the forward movement of the whole system in the medium is possible. The average speed of this movement over a period is determined. Optimal parameters of the motion of the internal mass for which the average speed of the system movement is maximal are found.     

The motion of a point mass on the surface of a smooth funnel

The motion of a point mass on a smooth concave surface (a funnel) under the action of a gravitational force is considered. The equations of motion are reduced to a form to which Lyapunov's theorem on the representation of the solution in the form of power series in the initial conditions, which converge absolutely in a finite region of phase space is applied. In the non-local formulation of the problem, a procedure is described for estimating the libration periods, based on an analysis of geometric forms. A bilateral estimate of the region of possible motion of the point is given for rotational-type motions, when the funnel is a surface of revolution.     

Multiplicity of Forced Oscillations on Manifolds and Applications to Motion Problems with One-Dimensional Constraints

This paper illustrates how the notion of an ejecting set recently introduced by the authors can be used to get multiplicity results for forced oscillations. The motion problem of a mass point constrained to a one-dimensional differentiable manifold and acted on by a periodic force is treated in detail.     

A special case of the motion of a point mass

When investigating the motion of a point mass in a plane. Zhukovskii [1] pointed out a case when, without finding the general integrals of the equations of motion, one can specify particular integrals of these motions. To obtain the particular integrals in explicit form, a certain constraint was imposed on the force function. The case of motion without this constraint is investigated.     

Drag force and virtual mass of a cylindrical porous shell in potential flow

We study the two-dimensional potential flow due to a circular cylinder in motion relative to an unbounded fluid. The cylinder consists of a thin, circular porous shell with fluid inside. The full nonlinear hydrodynamic problem is solved by Fourier expansion of Green's theorem. The truncated series is determined numerically by sampling points around the circle. A dimensionless shell parameter is introduced. For homogeneous porous shells, a maximal drag force occurs at the value 0.433 for the shell parameter, but the virtual mass is a monotonous function of the shell parameter. For an inhomogeneous shell, we have found a maximal value for the virtual mass which is 5% above the value for a rigid cylinder. Some of the results may be relevant to offshore engineering, especially in connection with porous coating of platform legs to reduce the total force.     

Numerical simulation of particles movement in cellular flows under the influence of magnetic forces

A numerical model of particle motion in fluid flow under the influence of hydrodynamic and magnetic forces is presented. As computational tool, a flow solver based on the Boundary Element Method is used. The Euler-Lagrange formulation of multiphase flow is considered. In the case of a particle with a magnetic moment in a nonuniform external magnetic field, the Kelvin body force acts on a single particle. The derived Lagrangian particle tracking algorithm is used for simulation of dilute suspensions of particles in viscous flows taking into account gravity, buoyancy, drag, pressure gradient, added mass and magnetophoretic force. As a benchmark test case the magnetite particle motion in cellular flow field of water is computed with and without the action of the magnetic force. The effect of the Kelvin force on particle motion and separation from the main flow is studied for a predefined magnetic field and different values of magnetic flux density. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transient vibration of an infinite string bearing a concentrated mass and supported by elastic-viscous suspensions

The transient transverse vibration of a string, bearing a concentrated mass and supported by elastic-viscous suspensions a constant distance apart, is investigated. The vibration is excited either by an instantaneous impulsive force or a constant force suddenly applied to the concentrated mass. In the first case a formal Fourier transformation of the Dirac impulse function is employed. In the second case the action of the constant force is regarded as a succession of actions of instantaneous pulses.     

Modelling and analysis of the nonlinear string-mass structure of the vibration absorber

ABSTRACT

In this paper a nonlinear string-mass structure of the vibration absorber is analyzed. This structure is convenient to be installed in vibration damping systems of high buildings for their protection in the case of earthquake. The considered string-mass structure contains a translator movable mass connected with two strings. Due to nonlinear geometric properties of the system the motion of the mass is described with a strong nonlinear second order differential equation. In the paper the approximate procedure for solving of the nonlinear equation of motion is developed. Based on the solution the influence of the string preloading force, slider mass and friction force on the vibration property of the string-mass system is investigated. It is concluded that variation of the preloading string force may be applied as a control parameter for vibration absorption and as the regulator of vibration decay time.     

Approximate equations of rotational motion of a rigid body carrying a movable point mass

The dynamics of a compound system, consisting of a rigid body and a point mass, which moves in a specified way along a curve, rigidly attached to the body is investigated. The system performs free motion in a uniform gravity field. Differential equations are derived which describe the rotation of the body about its centre of mass. In two special cases, which allow of the introduction of a small parameter, an approximate system of equations of motion is obtained using asymptotic methods. The accuracy with which the solutions of the approximate system approach the solutions of the exact equations of motion is indicated. In one case, it is assumed that the point mass has a mass that is small compared with the mass of the body, and performs rapid motion with respect to the rigid body. It is shown that in this case the approximate system is integrable. A number of special motions of the body, described by the approximate system, are indicated, and their stability is investigated. In the second case, no limitations are imposed on the mass of the point mass, but it is assumed that the relative motion of the point is rapid and occurs near a specified point of the body. It is shown that, in the approximate system, the motion of the rigid body about its centre of mass is Euler–Poinsot motion.     

On the stability of rods for stochastic excitations

The stability of motion of an elastic rod in a viscous medium compressed by a randomly acting force is studied. The conditions of stability of the rod acted upon by a stationary process with bilinear spectral density are obtained. The dependence of the statistical moments of the amplitude of the finite flexure of the rod under stationary-motion conditions on the parameters of the compressing force and the amplitude of the initial deformation is analysed. A number of problems concerning the stability of longitudinal flexure of viscoelastic constructions acted upon by random loads were discussed in /1–3/.     

Electromagnetic interaction of moving conductors with magnets: kinematic solutions for infinitely long square and cylindrical geometries

The electromagnetic drag force on a point dipole near a moving conductor caused by the induced electric currents is investigated by numerical and analytical computations. Our focus is on prototypical configurations for Lorentz force velocimetry, i.e. velocity measurement from the electromagnetic drag force on the dipole. We examine the particular cases of conducting infinite bars of square or round cross-section, which are moving with constant velocity in the field of arbitrary oriented magnetic dipole. In addition, we study the laminar liquid-metal flow in a square duct. The motion of the conductor is prescribed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonlinear fracture differential kinetic model to depict chaotic atom motions at a fatigue crack tip based on the differentiable manifold methodology

A nonlinear differential kinetic model describing dynamical behaviours of an atom at a fatigue crack tip is developed in this paper. It is assumed that the forces acted on this atom by its surrounding atoms consist of the following three components: (1) an elastic restoring force governed by Leonard-Jones potential, which describes the elastic interaction between atoms; (2) a nonlinear damping force proportional to its velocity through a linear function of its displacement as a coefficient that empirically simulates the energy loss from the crack tip to its surroundings; (3) an external remote driving force to represent thermally activated energy supplied to the crack tip from the surroundings. Based on these assumptions of the interaction forces between the atoms around the crack tip, a nonlinear dynamic equation describing the motion of the atom at a crack tip using the Newton’s second principle is derived. For a periodic external force and a random one influenced by parameters omitted, deterministic and a stochastic analyses on the dynamic equation obtained above are completed. Based on the theories of the Hopf bifurcation, global bifurcation and stochastic bifurcation, the extent and some possible implications of the existence of atomic-scale chaotic and stochastic bifurcative motions involving the fracture behaviour of actual materials are systematically and qualitatively discussed and the extreme sensitivity of chaotic motions to minute changes in initial conditions is explored. As demonstrated in the paper, chaotic behaviour may be observed in the case of a larger amplitude of the driving force and a smaller damping constant. The white noise introduced in the atomistic motion process may leads to a drift of the divergence point of the nonlinear stochastic differential kinetic system in contrast to the homoclinic divergence of the nonlinear deterministic differential kinetic system.     

Motion of a vibrating string in the presence of a convex obstacle: A free boundary problem

Here we study the motion of a vibrating string in the presence of an arbitrary obstacle. We show that if the string always rebounds on the concave parts of the obstacle, it can either rebound or roll on the convex parts. The latter is the case if the velocity of the string is null at the contact point just before contact, or if the contact point propagates at a characteristic speed. Four examples are given. The three first correspond to the same obstacle, a sinusoidal arc, but with different initial conditions. In the first case, the string rebounds on the whole of the obstacle and the motion is explicitly determined when it is periodic. In the second case, the string rolls on the convex part of the obstacle up to the inflexion point and then rebounds on the concave part and unwinds on the convex part. In the third case, the string is initially at rest on the obstacle; then it instantaneously leaves the concave part while it unwinds progressively on the convex part. The fourth case is similar to the third but with a different obstacle; the motion, which is periodic, is determined explicitly.     

Synthesis of a control in the problem of the time-optimal transfer of a point mass to a specified position with zero velocity

The problem of the time-optimal control of the motion of a point mass by means of a force of bounded modulus is considered. It is required that the point be transferred from an arbitrary state of motion to the origin of the system of coordinates with zero velocity. By introducing self-similar conjugate variables, the solution of the two-point problem can be successfully reduced to a search for the optimal root of a certain function, specified analytically. A complete solution of the control problem in the form of a synthesis is obtained using mathematical modelling methods. The feedback coefficients along the unit vectors of the position and velocity vectors are found and a control algorithm and a Bellman function are constructed. Examples using practical initial data are presented.