Eugene I. Butikov

On the dynamic stabilization of an inverted pendulum

A simple qualitative physical explanation is suggested for the phenomenon of dynamic stabilization of the inverted rigid planar pendulum whose pivot is constrained to oscillate with a high frequency in the vertical direction. A quantitative theory based on the suggested approach is developed. A computer program simulating the physical system supports the analytical investigation. The simulation reveals subtle details of the motion and aids the analytical study of the subject in a manner that is mutually reinforcing.

Parametric excitation of a linear oscillator

The phenomenon of parametric resonance is explained and investigated both analytically and with the help of a computer simulation. Parametric excitation is studied for the example of the rotary oscillations of a simple linear system— mechanical torsion spring pendulum excited by periodic variations of its moment of inertia. Conditions and characteristics of parametric resonance and regeneration are found and discussed in detail. Ranges of frequencies within which parametric excitation is possible are determined. Stationary oscillations at the boundaries of these ranges are investigated. The simulation experiments aid greatly an understanding of basic principles and peculiarities of parametric excitation and complement the analytical study of the subject in a manner that is mutually reinforcing.

The rigid pendulum - an antique but evergreen physical model

Various kinds of motion of a rigid pendulum (including swinging with arbitrarily large amplitudes and complete revolutions) are investigated both analytically and with the help of computerized simulations. The simulation experiments reveal many interesting peculiarities of this famous physical model and complement the analytical study of the subject in a manner that is mutually reinforcing.

Parametric resonance

Computer simulation experiments are especially good tools for helping students understand basic principles of physics. In this a, I have developed a package of simulation programs called Physics of Oscillations. One of its programs is suited for examining the phenomenon of paramagnetic resonance in a linear system. The program simulates the parametric excitation of the rotary oscillations of a mechanical torsion-spring pendulum whose moment of inertia is subject to periodic variations. I discuss the conditions and characteristics of parametric resonance, including paramagnetic regeneration. Instructors and their students can also use the Physics of Oscillations package to explore other problems, such as ranges of frequencies within which parametric excitation is possible and stationary oscillations on the boundaries of these ranges. The simulation experiments complement the analytical study of the subject in a manner that is mutually reinforcing.

Parametric resonance in a linear oscillator at square-wave modulation

The phenomenon of parametric resonance in a linear torsion spring oscillator caused by a square-wave modulation of its moment of inertia is explained and investigated both analytically and with the help of a computer simulation. Characteristics of parametric resonance and regeneration are found and discussed in detail. Ranges of frequencies within which parametric excitation is possible are determined. Stationary oscillations at the boundaries of these ranges and at the threshold conditions are investigated.

Subharmonic resonances of the parametrically driven pendulum

A simple qualitative physical explanation is suggested for the phenomenon of subharmonic resonances of a rigid planar pendulum whose axis is forced to oscillate with a high frequency in the vertical direction. An approximate quantitative theory based on the suggested approach is developed. The spectral composition of the subharmonic resonances is investigated quantitatively, and the boundaries of these modes in the parameter space are determined. New related modes of regular behaviour are described and explained. The conditions of the inverted pendulum stability are determined with greater precision than previous results. A computer program simulating the physical system supports the analytical investigation.

An improved criterion for Kapitza's pendulum stability

An enhanced and more exact criterion for dynamic stabilization of the parametrically driven inverted pendulum is obtained: the boundaries of stability are determined with greater precision and are valid in a wider region of the system parameters than previous results. The lower boundary of stability is associated with the phenomenon of subharmonic resonances in this system. The relationship of the upper limit of dynamic stabilization of the inverted pendulum with ordinary parametric resonance (i.e. with destabilization of the lower equilibrium position) is established. Computer simulation of the physical system aids the analytical investigation and proves the theoretical results.

Precession and nutation of a gyroscope

A simple treatment of the important old problem of the torque-induced rotation of a spinning symmetrical top is suggested. Our discussion is appropriate for teaching introductory mechanics and general physics to undergraduate students and is free from the difficulties of a traditional approach to the problem. The origin of nutation that accompanies forced precession is explained in detail, with an accent on its relationship to the torque-free precession of a symmetrical body. A small simulation program is developed that visualizes the investigated motion and illustrates its principal features. The program facilitates understanding about the counterintuitive behaviour of a gyroscope on a qualitative level.

A dynamical picture of the oceanic tides

A detailed treatment of tide-generating forces is given, followed by a simplified dynamic theory of tidal waves. To clarify the underlying physics, we use a simple model of the ocean that consists of a water shell of uniform depth completely covering the globe. The treatment is appropriate for college and university undergraduate students studying introductory geophysics or astronomy, general physics, or intermediate mechanics. A computer simulation is developed to aid in understanding the properties of sun- or moon-induced tide-generating forces and of the stationary tidal waves created by these forces in the open ocean.

Inertial rotation of a rigid body

A simple approach to the important problem of torque-free rotation of a symmetrical rigid body is suggested which is appropriate for teaching introductory mechanics and general physics to undergraduate students and is free from the difficulties of traditional treatment of the problem. A small simulation program (Java-applet) is developed that visualizes the investigated motion and illustrates its principal features. The program facilitates understanding of concepts behind rigid body dynamics. Simultaneously with simulating the rigid body motion, the program presents a clear geometrical interpretation of the inertial rotation.