Dynamic Behavioral Analysis of an Antiseismic Support under an External Kinematic Disturbance

Study is made of the dynamic behavior of an antiseismic support, which is a heavy homogeneous ball rolling within a cycloidal cavity between mobile upper and lower slabs, subjected to an external kinematic disturbance. The equations of motion are derived for the heavy ball moving in the cycloidal cavity without slipping, and the dynamic effects in this, new earthquake-protective system are analyzed numerically.     

On stability of relative equilibria of a double pendulum with vibrating suspension point

We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical. The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting [1–3] and a nonlinear setting [4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum [6–9] and a double pendulum [10, 11]. In [12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting.     

Rigid body impact with moment of rolling friction

In this paper, the impact between a rigid pendulum and rough surfaces is studied. The rolling friction moment and the coefficient of rolling friction are introduced, and an improved mathematical model of the planar impact with friction is presented. The influence of the moment of rolling friction on the energy dissipated by friction during the impact is analyzed. For a simple pendulum, using the energetic coefficient of restitution, more energy is dissipated for larger values of the coefficient of kinetic friction and contact radius, and for smaller values of the length of the beam. For a double pendulum using the kinematic coefficient of restitution, some energetically inconsistent results can be solved for some values of the coefficient of rolling friction.     

Vibration energy harvesting for low frequency using auto-tuning parametric rolling pendulum under exogenous multi-frequency excitations

An electromagnetic parametrically excited rolling pendulum energy harvester with self-tuning mechanisms subject to multi-frequency excitation is proposed and investigated in this paper. The system consists of two uncoupled rolling pendulum. The resonance frequency of each the rolling pendulum can be automatically tuned by adjusting its geometric parameters to access parametric resonance. This harvester can be used to harvest the energy at low frequency. A prototype is developed and evaluated. Its mathematical model is derived. A cam with rolling follower mechanism is employed to generate multi-frequency excitation. An experimental study is conducted to validate the proposed concept. The experimental results are confirmed by the numerical results. The harvester is successfully tuned when the angular velocity of the cam is changed from 1.149 to 1.236 Hz.     

WKB method for analyzing the vibrations of a Meshcherskii pendulum

TheWKB method is used to construct an approximate analytic solution of the equation of small nonstationary vibrations of the Meshcherskii mathematical pendulum. The versions of linear and nonlinear variation in the pendulum mass are considered. Calculations showed that, under certain restrictions on the pendulum parameters, the approximate solutions constructed in elementary functions are a good approximation to the exact results.     

Rotation of the apparent vibration plane of a swinging spring at the 1:1:2 resonance

Nonlinear spatial vibrations of a mass point on a weightless elastic suspension (pendulum on a spring) are considered. The frequency of vertical vibrations is assumed to be equal to the doubled swinging frequency (the 1:1:2 resonance). In this case, as numerical calculations and experiments show, the vertical vibrations are unstable, which leads to the vertical vibration energy transfer to the pendulum swinging energy. The vertical vibrations of the mass point decay and, after a certain time period, the pendulum starts swinging in a certain vertical plane. This swinging is also unstable, which results in the reverse energy transfer into the vertical vibration mode. The vertical vibrations are again repeated. But after the second transfer of the vertical vibration energy to the pendulum swinging energy, the apparent plane of vibrations rotates by a certain angle. These effects are described analytically; namely, the energy transfer period, the time variations in the amplitudes of both modes, and the variations in the angle of the apparent vibration plane are determined. An asymptotic solution is also constructed for the mass point trajectory in the orbit elements. In projection on the horizonal plane, the mass point moves in a nearly elliptic trajectory. The ellipse semiaxes slowly vary with time, so that their product remains constant, and the major semiaxis slowly rotates at a constant sectorial velocity. The obtained analytic time dependence of the ellipse semiaxes and the precession angle agree well with the results of numerical calculations.     

Three‐dimensional computation for flow‐induced vibrations in in‐line and cross‐flow directions of a circular cylinder

We present numerical results for in‐line and cross‐flow vibrations of a circular cylinder, which is immersed in a uniform flow and is elastically supported by damper‐spring systems to compute vibrations of a rigid cylinder. In the case of a circular cylinder with a low Scruton number, it is well‐known that two types of self‐excited vibrations appear in the in‐line direction in the range of low reduced velocities. On the other hand, a cross‐flow vibration of the circular cylinder can be excited in the range of high reduced velocities. Therefore, we compute the flow‐induced vibrations of the circular cylinder in the wide range of the reduced velocities at low and high Scruton numbers and discuss about excitation mechanisms in the in‐line and cross‐flow directions. Copyright © 2011 John Wiley & Sons, Ltd.     

Vibrations of the elasto-plastic pendulum

A numerical strategy for vibrations of elasto-plastic beams with rigid-body degrees-of-freedom is presented. Beams vibrating in the small-strain regime are considered. Special emphasis is laid upon the development of plastic zones. An elasto-plastic beam performing plane rotatory motions about a fixed hinged end is used as example problem. Emphasis is laid upon the coupling between the vibrations and the rigid body rotation of the pendulum. Plastic strains are treated as eigenstrains acting in the elastic background structure. The formulation leads to a non-linear system of differential algebraic equations which is solved by means of the Runge-Kutta midpoint rule. A low dimension of this system is obtained by splitting the flexural vibrations into a quasi-static and a dynamic part. Plastic strains are computed by means of an iterative procedure tailored for the Runge-Kutta midpoint rule. The numerical results demonstrate the decay of the vibration amplitude due to plasticity and the development of plastic zones. The pendulum approaches a state of plastic shake-down after sufficient time.     

On the motions of a double pendulum with vibrating suspension point

We consider the motions of a system consisting of two pivotally connected physical pendulums rotating about horizontal axes. We assume that the system suspension point, which coincides with the suspension point of one of the pendulums, performs harmonic vibrations of high frequency and small amplitude along the vertical. We also assume that the system has four relative equilibrium positions in which the suspension points and the pendulum centers of mass lie on one vertical line. We study the stability of these relative equilibria. For arbitrary physical pendulums, we obtain stability conditions in the linear approximation. For a system consisting of two identical rods, we solve the stability problem the in nonlinear setting. For the same system, we study the existence, bifurcations, and stability of high-frequency periodic motions of small amplitude other than the relative equilibria on the vertical line. The studies of dynamic stability augmentation in mechanical systems under the action of high-frequency perturbations was initiated in the paper [1], where it was shown that the unstable inverted equilibrium of a pendulum may become stable if the suspension point vibrates rapidly. This idea was developed in [2–10] and other papers, where several aspects of motion of a mathematical pendulum in the case of rapid small-amplitude vibrations of the suspension point were studied in the linear setting and also (without full mathematical rigor) in the nonlinear setting. The motions of the suspension point along an arbitrary oblique straight line [2, 4, 7, 8], along the vertical [3, 5, 6], along the horizontal [9], and in the case of damping [8] were considered. The monograph [10] deals with the stabilization of a pendulum or a system of pendulums under periodic and conditionally periodic vibrations of the suspension point along the vertical, along an oblique straight line, and along an ellipse. A rigorous nonlinear analysis of the existence and stability of periodic motions of the mathematical pendulum under horizontal and oblique vibrations of the suspension point at arbitrary frequencies and amplitudes can be found in [11, 12]. For the case of vertical vibrations of the suspension point at an arbitrary frequency and amplitude, a rigorous stability analysis of the relative equilibria of the pendulum on the vertical was carried out in [13].     

Nonlinear design analysis of centrifugal pendulum vibration absorbers: an intrinsic geometry-based framework

Nonlinear Dynamics - The centrifugal pendulum vibration absorber (CPVA) is a device whose purpose is the reduction in torsional vibrations in rotating and reciprocating machinery. Over the last...     

Two classes of vibrations of a short circular cylinder

This paper gives a solution of the stationary dynamic problem of elasticity which describes two classes of natural nonaxisymmetric vibrations of a finite circular cylinder. In the particular case of axial symmetry, the resulting solution describes two well-known classes of axisymmetric vibrations: vibrations of the first class become longitudinal-transverse vibrations and vibrations of the second class become torsional vibrations. The existence of two classes of nonaxisymmetric vibrations is due to the boundary conditions at the ends. It is shown that as the length (height) of the cylinder increases, the effect of the boundary conditions at the ends on the frequency spectrum reduces, and the vibration frequencies of the two classes become similar and then identical.     

Manifestation of Secondary Resonances during Pendulum Vibrations of a Tankwith Fluid

International Applied Mechanics - The problem of vibrations of a tank on pendulum suspension with a movable suspension point and partially filled with an ideal incompressible fluid is considered....     

Free v/s forced vibrations of a cylinder at low Re

Free vibrations of a circular cylinder of low non-dimensional mass are investigated at low Reynolds numbers. Computations are carried out for 5% blockage. Lock-in is observed for a range of Re and is accompanied with hysteresis at both lower as well as higher Re ends of the synchronisation/lock-in region. It is well known that the lock-in regime for free vibrations depends on the non-dimensional mass of the oscillator. The results from the present computations are compared with the data for forced vibrations from Koopmann (Journal of Fluid Mechanics, 28, 501–512, 1967) on a Y _max/D vs. f* plot, where Y _max is the maximum oscillation amplitude and f* is the ratio of cylinder vibration frequency to the vortex shedding frequency for a stationary cylinder. Good agreement is observed for the critical amplitude needed for onset of synchronisation between the forced and free vibrations. The results from the free vibrations are compared to the predictions from the linear oscillator model by assuming that the forces on the cylinder are unaffected as a result of vibrations. It is found that, for low mass oscillators, the modification of vortex shedding frequency and lift coefficient due to cylinder oscillations leads to the enhancement of the lock-in regime.     

Flow control in flow–structure interaction

We employ passive flow control using two-dimensional hydrofoils to reduce vortex-induced vibrations (VIV) and drag on a cylinder of circular cross-section. We test the hypothesis that by using foils to bend the streamlines around the cylinder, and hence forcing the flow to approach potential flow-like patterns VIV and drag will be reduced. A systematic parametric search, first using groups of two and then four foils, shows that it is possible to completely eliminate vibrations and reduce the drag coefficient to about C_d=0.50 at sub-critical Reynolds numbers. This parametric search is conducted in conjunction with force measurement and particle image velocimetry on a fixed towed cylinder. The effectiveness of the foils in regards to VIV was further tested with an apparatus allowing free transverse vibrations of a towed cylinder.     

The use of impulse response functions for evaluation of added mass and damping coefficient of a circular cylinder oscillating in linearly stratified fluid

 The damped horizontal oscillations of a circular cylinder in linearly stratified fluid are studied experimentally. The cylinder is fixed to the lower end of a physical pendulum with variable restoring moment. The impulse response function of the pendulum in time domain is recorded and converted to the frequency response function using Fourier transform. The density stratification is shown to have a strong effect on frequency-dependent hydrodynamic coefficients (added mass and damping). The data obtained are compared with available theoretical predictions. The applicability of a simplicistic method implying approximation of impulse response functions by analytical functions is discussed. Received: 15 September 1998/Accepted: 1 April 1999     

Vibrations of a cylinder in a uniform flow in the presence of a no-slip side-wall

A circular cylinder placed in a uniform flow, and that spans the entire length between two side walls, may experience either parallel or oblique vortex shedding depending on the end conditions. It was shown by Mittal and Sidharth (2014) that the spatio-temporal periodicity of the oblique vortex shedding results in constant-in-time force experienced by the cylinder. On the contrary, parallel vortex shedding leads to fluid force that fluctuates with time. The free vibrations of a circular cylinder, in the presence of a wall, are investigated. For comparison, computations with end walls, where a slip condition on velocity is specified, are also carried out. The Reynolds number, based on the diameter of the cylinder and free-stream speed of the flow, is Re=100. The initial condition for the free vibrations is the fully developed unsteady flow past a stationary cylinder with oblique shedding. It is found that as the amplitude of vibration of the cylinder builds up, the vortices shed from the cylinder align with its axis leading to parallel shedding. The response of the cylinder is associated with two branches: initial and lower. On the lower branch, the response of the cylinder is virtually identical from two- and three-dimensional computations. The flow as well as the response is different on the initial branch and outside the synchronization regime. Forced vibrations confirm the phenomena.     

Brachistochrone for a rolling cylinder

The motion of a heavy homogeneous cylinder is considered as a no-slip rolling along the desired curve. We obtain a functional in the form of the total time of the cylinder rolling and solve the corresponding variational problem of minimizing this functional. We obtain an algebraic equation for the directional line of steepest descent, brachistochrone, in parametric form. We use the equation of motion of the cylinder with constraint reaction to determine the conditions of implementation of its pure rolling without separation and slip with respect to the brachistochrone.     

Estimation of parameters of various damping models in planar motion of a pendulum

Meccanica - In this work, planar free vibrations of a single physical pendulum are investigated both experimentally and numerically. The laboratory experiments are performed with pendula of...     

On Spatial Behavior of the Harmonic Vibrations in Kelvin-Voigt Materials

The present paper deals with the study of the amplitude of the steady-state vibrations in a right finite cylinder made of an isotropic Kelvin-Voigt material. Some exponential decay estimates, similar to those of Saint-Venant type, are obtained for appropriate cross-sectional area measures associated with the amplitude of the steady-state vibrations. It is proved that due to dissipative effects, the estimates in question hold for every value of the frequency of vibrations and for arbitrary values of the elastic coefficients. The results are extended to a semi-infinite cylinder and some alternatives of Phragmèn-Lindelöf type are established.