Elimination of tall building vibration resulting from ground motions

In this paper, the vibration problems of tall buildings are considered. The focus is on vibration caused by earthquakes, semi–seismic phenomena and ground vibrations of other origins. The construction consists of the main system and a vibration eliminator (passive tuned mass damper – pendulum type) which is attuned to the first eigenfrequency of the main structure. The analysis focuses on elimination of structure vibration caused by horizontal components of ground motions, while the functioning of the eliminator is simultaneously influenced by the vertical component (parametric effect – the possibility of improper functioning of the device). The vertical periodic movement of the support point can cause changes of the vibration eliminator's stiffness. In such a case parametric excitation occurs in the system, which signifies that parametric resonance may appear. The numerical analysis of the problem was performed with the Newmark method in conjunction with FEM. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cauchy problem for the Mathieu equation away from parametric resonance

Four solutions of the Cauchy problem for Mathieu’s equation away from parametric resonance domains are analytically constructed using an asymptotic averaging method in the fourth approximation. Three solutions occur near fractional parameter values at which slow combination phases exist. The fourth solution occurs in the absence of slow phases away from parametric resonance domains and the fractional parameter values.     

Non-periodic and chaotic vibrations in a flow induced system

A flow induced system, consisting of an elastically mounted body with a pendulum attached, is considered here. The stability of the semi-trivial solution, representing the vibration of the body with the non-oscillating pendulum, is investigated. The analytical investigation shows that at a certain flow velocity, higher than the critical one, the pendulum begins to oscillate due to autoparametric resonance. For a convenient tuning, the vibration of the system can be substantially reduced. The analysis of both semi-trivial and non-trivial solutions is complemented by numerical integration of the differential equations of motion. A mapping technique based on Poincaré section, suitable for investigating the non-periodic vibrations occuring at higher flow velocities, is proposed.     

Stick balancing with reflex delay in case of parametric forcing

The effect of parametric forcing on a PD control of an inverted pendulum is analyzed in the presence of feedback delay. The stability of the time-periodic and time-delayed system is determined numerically using the first-order semi-discretization method in the 5-dimensional parameter space of the pendulum’s length, the forcing frequency, the forcing amplitude, the proportional and the differential gains. It is shown that the critical length of the pendulum (that can just be balanced against the time-delay) can significantly be decreased by parametric forcing even if the maximum forcing acceleration is limited. The numerical analysis showed that the critical stick length about 30 cm corresponding to the unforced system with reflex delay 0.1 s can be decreased to 18 cm with keeping maximum acceleration below the gravitational acceleration.     

A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

Parametric bufferness in systems of parabolic and hyperbolic equations with small diffusion

We investigate the problem of parametric excitation of oscillations in systems of parabolic and hyperbolic equations with small coefficient of diffusion. We establish the phenomenon of parametric bufferness, i.e., the existence of an arbitrary fixed number of stable periodic solutions for a proper choice of the parameters of equations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 22–35, January, 1998.     

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)     

Parametric Integer Programming Algorithm for Bilevel Mixed Integer Programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader’s variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case, it yields a “better than fully polynomial time” approximation scheme with running time polynomial in the logarithm of the absolute precision. For the pure integer case where the leader’s variables are integer, and hence optimal solutions are guaranteed to exist, we present an algorithm which runs in polynomial time when the total number of variables is fixed.     

Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness

In this paper, we investigate the Shilnikov type multi-pulse chaotic dynamics for a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time-varying in a periodic form. The dimensionless equation of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions is a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found from the numerical results that there are the phenomena of the Shilnikov type multi-pulse chaotic motions for the rotor-AMB system. A new jumping phenomenon is discovered in the rotor-AMB system with the time-varying stiffness.     

The stability of the equilibrium of a pendulum of variable length

The frequencies and modes of parametric oscillations of a pendulum of variable length for values of the modulation index from the smallest to the limit admissible values are investigated. The limits of the resonance zones of the first four oscillation modes are constructed and investigated by analytical and numerical methods, and the fundamental qualitative properties of the higher modes are established. Complete degeneracy of the modes with even numbers, i.e., coincidence of the frequencies of the odd and even eigenmodes for admissible values of the modulation parameter, is proved. The global pattern of the limits of the regions of stability of the lower position of equilibrium is constructed and it is shown that it differs considerably from the Ince–Strutt diagrams. Specific properties of the eigenmodes are established.     

Coherentization of the energy of heat fluctuations by a two-channel bilinear control system

We propose and investigate a mathematical model of an open bilinear control system for the conversion of heat energy in a coherent form. We show that the use of a combinational parametric resonance formed by the control system in a one-temperature ensemble of weakly dissipative elastic-gyroscopic subsystems enables one to obtain a positive energy output without using any cooling device apart from the control system. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1557–1573, November, 2007.     

Observation and investigation of a non-monotonic period function for an impact oscillator

A mathematical model for a bench-top impact oscillator (a steel pendulum and an aluminum barrier) that incorporates Hertzian contact is analyzed, and predictions derived from our model are compared with experimental data. We report our discovery of a new effect: the existence of a non-monotone period function for the annulus of periodic orbits in the phase plane surrounding the rest point that corresponds to the downward vertical position of the pendulum in the unforced, undamped case. From this non-monotonicity, we predict novel resonance response behavior for the forced, damped oscillator and verify our predictions by experiment.     

On the application of Ateb-functions to the construction of a solution of a nonlinear Klein-Gordon equation

For a nonlinear Klein-Gordon equation, we construct the first approximation of an asymptotic solution by using Ateb-functions. The resonance and nonresonance cases are considered. “L’vivs’ka Politeknika” University, Lviv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 6, pp. 872–877, June, 1997.     

Algorithm for constructing an optimal control strategy in a nonlinear differential game with Lipschitz compactly supported payoff

For a zero-sum differential game, we consider an algorithm for constructing optimal control strategies with the use of backward minimax constructions. The dynamics of the game is not necessarily linear, the players’ controls satisfy geometric constraints, and the terminal payoff function satisfies the Lipschitz condition and is compactly supported. The game value function is computed by multilinear interpolation of grid functions. We show that the algorithm error can be arbitrarily small if the discretization step in time is sufficiently small and the discretization step in the state space has a higher smallness order than the time discretization step. We show that the algorithm can be used for differential games with a terminal set. We present the results of computations for a problem of conflict control of a nonlinear pendulum.     

Crane Dynamics with Modulated Hoisting

For many types of cranes commonly used in technical applications, the reduction of payload pendulations is an important design issue. Especially for cranes with variable cable length, oscillations are boosted by the hoisting of the payload due to nonlinear effects. Most of the techniques for active damping are based on a control input that displaces the support of the hoisting mechanism perpendicularly to the direction of the pendulum. However, controlled motion of the carrying structure might not be suitable or even impossible for some applications. The possibility to influence and reduce pendulations by means of feedback controlled variations of the cable length is hardly used in crane technology. A control strategy based on the phenomenon of autoparametric resonances in nonlinear dynamical systems is presented that manipulates the desired hoisting velocity by superposition of a suitably modulated motion in order to reduce amplifications of the pendulations, in particular in absence of other effective control inputs. Experimental results for a simple pendulum setup are presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rotating orbits of a parametrically-excited pendulum

The authors consider the dynamics of the harmonically excited parametric pendulum when it exhibits rotational orbits. Assuming no damping and small angle oscillations, this system can be simplified to the Mathieu equation in which stability is important in investigating the rotational behaviour. Analytical and numerical analysis techniques are employed to explore the dynamic responses to different parameters and initial conditions. Particularly, the parameter space, bifurcation diagram, basin of attraction and time history are used to explore the stability of the rotational orbits. A series of resonance tongues are distributed along the non-dimensionalied frequency axis in the parameter space plots. Different kinds of rotations, together with oscillations and chaos, are found to be located in regions within the resonance tongues.     

The high-temperature relaxation function of a spin system with a quadratic contribution of fluctuations to the resonance frequency

We investigate the high-temperature relaxation function of a spin system with quadratic coupling of the resonance frequency to the Gaussian random process. In the general case, this function is expressed as an integral of an infinite auxiliary series. For theN-exponential Gauss Markov process, the problem is reduced to solving a system of 2N linear equations. For brevity, we analyze the effect of fluctuations on the form of the magnetic resonance line (the Fourier image of the relaxation function). For both the one- and multiexponential processes in a crystal with dynamics of a relaxation type in the continuous phase transition domain, we find a nonmonotonic dependence of the asymmetrical homogeneously widened resonance line on the rate of fluctuations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 316–328, November, 1999.     

Control of the spatial motion of a multilink inverted pendulum using a torque applied to the first link

An approach to constructing a control for non-linear mechanical systems in which the number of degrees of freedom exceeds the dimension of the generalized control forces is developed. An n-link pendulum with two-degree-of-freedom joints, controlled by a torque applied to the first link, is considered as an example. Such a pendulum has 2ⁿ different equilibrium positions. A feedback control with an absolute value constraint, which transfers the pendulum from the neighbourhood of an arbitrary equilibrium position to this equilibrium position in a finite time, is constructed. For this purpose, the equations of motion of the pendulum are linearized in the neighbourhood of the equilibrium position under consideration, complete controllability of the linear model is established, and a control is constructed for it using the linear matrix inequality technique. The applicability of the control law obtained to the solution of the problem of controlling a non-linear multilink pendulum is verified. ©2014     

Waggling conical pendulums and visualization in mechanics

Students often find mechanics a difficult area to grasp. This paper describes an equation of motion for a waggling conical pendulum. A wide range of pendulum dynamics can be simulated with this model. The equation of motion is embedded in a graphical user interface (GUI) for its numerical solution in MATLAB. This allows a student's focus to be on the influence of different parameters on the pendulums dynamics. The simulation tool can be used as a dynamics demonstrator in a lecture or as an educational tool driven by the imagination of the student. By way of demonstration, the simulation tool has been applied to two damped pendulums and an inverted damped pendulum. The model has also been used to simulate resonance and has shown that there is a wide range of behaviour possible depending on the type of forcing applied. Finally, a forced conical pendulum as a system for harnessing wave energy is considered.