Stability condition for vertical oscillation of 3-dim heavy spring elastic pendulum

Equations of motion for 3-dim heavy spring elastic pendulum are derived and rescaled to contain a single parameter. Condition for the stability of vertical large amplitude oscillations is derived analytically relating the parameter of the system and the amplitude of the vertical oscillation. Numerical continuation is used to find the border of the stability region in parameter space with high precision. The stability condition is approximated by a simple formula valid for a large range of the parameter and of the amplitude of oscillation. The bifurcation responsible for the loss of stability is identified.      

A magnet spring model

The paper discusses an elementary spring model representing the motion of a magnet suspended from the ceiling at one end of a vertical spring which is held directly above a second magnet fixed on the floor. There are two cases depending upon the north–south pole orientation of the two magnets. The attraction or repelling force induced by the magnets follow an inverse quartic law and thus we are led to a nonlinear model suitable for discussion in a beginning differential equations course. Spring models are common fare in such courses, but usually only linear models with simple sinusoidal forcing are considered. The resultant model is autonomous and thus an energy approach permits a full phase portrait of the resultant motions in the phase plane. These phase portraits show interesting behaviour of the system, reinforcing one's natural physical intuition. The computer algebra system Mathematica is employed here, although almost any other system would suffice. Such a system permits almost effortless calculations and can generate the graphics needed to thoroughly investigate the model.     

Studying Dynamical Principles of Human Locomotion using Inverse Optimal Control

Human movement, as for example human gait, can be considered as an optimal realization of some given task. However, the criterion for which the naturally performed human motion is optimal, is generally not known. In this article we formulate an inverse optimal control problem to study the relevance of four different optimization criteria in human locomotion. As a walking model we use an actuated three dimensional spring loaded inverted pendulum (3D-SLIP), which is able to mirror the typical shape of the center of mass trajectory in human gait. Using a direct all-at-once approach, the weighting of the optimization criteria and the position of the footsteps are optimized in such a way, that the center of mass trajectory of the resulting optimal state fits real motion capture data as good as possible. Numerical experiments show, that whereas the so called capture point seems to have a great impact on human walking, minimization of the vertical center of mass movement does not show any relevance at all. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On loss of stability of slow motions in stiff oscillatory systems

In stiff oscillatory systems often a reduction of the order of the system is possible by splitting the motion into an essential motion on a nearby slow manifold and neglecting the fast motion. However, if the system is conservative the question of stability of the slow motion is a delicate problem. For various spring pendulum systems we, first, perform numerical simulations showing that if the stiffness of the springs is gradually reduced the slow motion looses stability. For a single spring pendulum we give an explanation of this loss of stability. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hydromagnetic stability of convective flow of variable viscosity fluids generated by internal heat sources

The stability of convective motion of a variable viscosity fluid contained in a vertical layer generated by uniformly distributed internal heat sources in the presence of a transverse magnetic field is studied. The viscosity of the fluid is assumed to depend on the temperature. The undisturbed steady state motion is assumed to consist of purely vertical motion with a nonlinear temperature distribution across the layer. The equations were solved by the spectral collocation method. The results show that thermal running waves are the most unstable modes and dominate the shear modes when the viscosity decreases.     

Hydromagnetic stability of convective flow of variable viscosity fluids generated by internal heat sources

The stability of convective motion of a variable viscosity fluid contained in a vertical layer generated by uniformly distributed internal heat sources in the presence of a transverse magnetic field is studied. The viscosity of the fluid is assumed to depend on the temperature. The undisturbed steady state motion is assumed to consist of purely vertical motion with a nonlinear temperature distribution across the layer. The equations were solved by the spectral collocation method. The results show that thermal running waves are the most unstable modes and dominate the shear modes when the viscosity decreases.     

Bifurcation of slow motions in a stiff spring pendulum

We consider free oscillations of a double pendulum, where one of the pendula is modelled as a very stiff spring. Contrary to a single spring pendulum numerical simulations show an unexpected large influence of the fast longitudinal oscillations on the slow pendulum oscillations even for extremely large values of the stiffness. The transition from the regular motion, which is governed by the dynamics of a rigid double pendulum close to a periodic orbit, to the irregular motion with large contributions from the longitudinal oscillations occurs due to a subcritical symmetry breaking bifurcation of the periodic solution. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The dynamics of a spherical pendulum with a vibrating suspension

The motion of a spherical pendulum whose point of suspension performs high-frequency vertical harmonic oscillations of small amplitude is investigated. It is shown that two types of motion of the pendulum exist when it performs high-frequency oscillations close to conical motions, for which the pendulum makes a constant angle with the vertical and rotates around it with constant angular velocity. For the motions of the first and second types the centre of gravity of the pendulum is situated below and above the point of suspension, respectively. A bifurcation curve is obtained, which divides the plane of the parameters of the problem into two regions. In one of these only the first type of motion can exist, while in the other, in addition to the first type of motion, there are two motions of the second type. The problem of the stability of these motion of the pendulum, close to conical, is solved. It is shown that the first type of motion is stable, while of the second type of motion, only the motion with the higher position of the centre of gravity is stable.     

Modelling and simulation of a sphere rolling on the inside of a rough vertical cylinder

A classical problem of nonholonomic system dynamics—the motion of a sphere on the inside of a rough vertical cylinder—is extended to rolling friction. The case study is modelled in independent coordinates. Due to the nonholonomic constraints imposed on the sphere, the governing equations arise as a set of differential-algebraic equations. The results of numerical simulations show the transition of the sphere from a sinusoid path on the vertical cylinder surface to a fall with slip. The physics of the ‘paradoxical’ motion is explained in detail.     

On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane

A material system consisting of an outer rigid body (a shell) and an inner body (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its linear stability is studied.     

The stability of the equilibrium of a pendulum for vertical oscillations of the point of suspension

The motion of a pendulum, the point of suspension of which is subject to vertical harmonic oscillations of arbitrary frequency and amplitude, is considered. A complete rigorous solution of the non-linear problem of the stability of the relative positions of equilibrium of the pendulum along the vertical is given.     

Analysis of Vertebrae Motion with Computer Simulation Using Elementary Contact Pairs

To simulate the motion of two adjacent vertebrae a computer model is created. Described in the paper is the mathematical modelling of a vertebrae pair using multibody methods and impact analysis techniques with elementary contact geometry for the facet joints. The ligaments and the intervertebral disc are modeled as a parallel connection of a spring and a damper. The results are compared with existing approaches and experimental data. In order to enhance the results, piecewise linear functions instead of a constant spring stiffness for the model of the intervertebral disc are used. The simulation allows the motion in three-dimensional space (three translations and three rotations, see figure 1). The investigations are focused on the vertebrae pair C5-C6 but can be easily extended to other vertebrae. (© 2005 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.     

Motion of a Swinging Atwood’s Machine: Simulation and Analysis with Mathematica

A generalized model of the Atwood machine when one body is constrained to move along a vertical axis while the other one can swing in a plane is considered. Combining symbolic and numerical calculations, we have obtained equations of motion of the system and analyzed their solutions. We have shown that oscillation can completely modify a motion of the system while the simple Atwood machine demonstrates only the uniformly accelerated motion of the bodies. The validity of the results obtained is demonstrated by means of the simulation of motion of swinging Atwood’s machine with the computer algebra system Wolfram Mathematica.     

On Head Seas Travelling Along a Horizontal Cylinder

Consider a wave train of arbitrary wavelength travelling withoutchange of form along a partially immersed fixed horizontal cylinder,the wave crests being normal to the generators of the cylinder.It is supposed that the cylinder is symmetrical about its longitudinalmid-plane, and that the wave motion is also symmetrical aboutthis plane. At a distance from the cylinder the motion is supposedto approximate to the incident wave train. This wave motionis a limiting form of the motion near a long ship in head seas.It is the purpose of the present work to show that under theusual assumptions of linearized wave theory there can be nosuch wave motion. In other words, according to the linearizedtheory a head sea must deform as it travels along a horizontalcylinder. (The proof fails for those wavelengths, if any, forwhich the Fredholm determinant of a certain integral equationvanishes. There is as yet no general uniqueness theory withoutsuch a limitation.) To illustrate this conclusion a particular problem is treated,corresponding to a head sea travelling along a wall which isslightly inclined to the vertical along part of its length andis exactly vertical elsewhere. For this case the progressivedeformation can be calculated.     

Fuzzy multi-objective optimization of passive suspension parameters

The paper proposed a systematic and effective optimization process to optimize a 3-D vehicle suspension dynamic model with eight DOF, including seat vertical motion, suspension vertical, pitching and rolling motions and wheels vertical motions using fuzzy optimization, to attain the best compromise between ride comfort and vehicle-generated road damage. The results show a substantial improvement in the vertical ride quality is obtained while keeping the suspension deflections within their allowable clearance when the vehicle moves at a constant velocity v = 20m/s, and the comfort performance of a suspension seat can be considerably enhanced.     

Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator

We investigate the interaction effect of fast vertical parametric excitation and time delay on self-oscillation in a van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic and then we apply the averaging technique on this slow dynamic to derive a slow flow. In particular we analyze the slow flow to analytically approximate regions where self-excited vibrations can be eliminated. Numerical integration is performed and compared to the analytical results showing a good agreement for small time delay. It was shown that vertical parametric excitation, in the presence of delay, can suppress self-excited vibrations. These vibrations, however, persist for all values of the excitation frequency in the case of a fast vertical parametric excitation without delay [Bourkha R, Belhaq M. Effect of fast harmonic excitation on a self-excited motion in van der Pol oscillator. Chaos, Solitons & Fractals, 2007;34(2):621–7.].     

Nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam, additionally supported by a nonlinear spring

The free and forced vibrations of a Kelvin-Voigt viscoelastic beam, supported by a nonlinear spring are analytically investigated in this paper. The governing equations of motion along with the compatibility conditions are obtained employing Newton’s second law of motion and constitutive relations. The viscoelastic beam material is constituted by the Kelvin-Voigt rheological model, which is a two-parameter energy dissipation model. The method of multiple timescales, a perturbation technique, is employed which ultimately leads to approximate analytical expressions for vibration response, and provides better insight into how the system parameters influence the vibration response. Finally, the effect of system parameters on the linear and nonlinear natural frequencies, vibration responses and frequency-response curves of the system is characterized.     

Control of an aircraft landing in windshear

The problem of the feedback control of an aircraft landing in the presence of windshear is considered. The landing process is investigated up to the time when the runway threshold is reached. It is assumed that the bounds on the wind velocity deviations from some nominal values are known, while information about the windshear location and wind velocity distribution in the windshear zone is absent. The methods of differential game theory are employed for the control synthesis.The complete system of aircraft dynamic equations is linearized with respect to the nominal motion. The resulting linear system is decomposed into subsystems describing the vertical (longitudinal) motion and lateral motion. For each subsystem, an, auxiliary antagonistic differential game with fixed terminal time and convex payoff function depending on two components of the state vector is formulated. For the longitudinal motion, these components are the vertical deviation of the aircraft from the glide path and its time derivative; for the lateral motion, these components are the lateral deviation and its time derivative. The first player (pilot) chooses the control variables so as to minimize the payoff function; the interest of the second player (nature) in choosing the wind disturbance is just opposite.The linear differential games are solved on a digital computer with the help of corresponding numerical methods. In particular, the optimal (minimax) strategy is obtained for the first player. The optimal control is specified by means of switch surfaces having a simple structure. The minimax control designed via the auxiliary differential game problems is employed in connection with the complete nonlinear system of dynamical equations.The aircraft flight through the wind downburst zone is simulated, and three different downburst models are used. The aircraft trajectories obtained via the minimax control are essentially better than those obtained by traditional autopilot methods.     

The equations of the approximate theory of the motion of a rigid body with a vibrating suspension point

The motion of a rigid body in a uniform gravity field is investigated. One of the points of the body (the suspension point) performs specified small-amplitude high-frequency periodic or conditionally periodic oscillations (vibrations). The geometry of the body mass is arbitrary. An approximate system of differential equations is obtained, which does not contain the time explicitly and describes the rotational motion of the rigid body with respect to a system of coordinates moving translationally together with the suspension point. The error with which the solutions of the approximate system approximate to the solution of the exact system of equations of motion is indicated. The problem of the stability with respect to the equilibrium of the rigid body, when the suspension point performs vibrations along the vertical, is considered as an application.