TY - CHAP

T1 - Examples

AU - Fradkov, Alexander L.

PY - 2007

Y1 - 2007

N2 - In the beginning of the 20th century, Scottish professor P. Stephenson has shown mathematically that the upper unstable equilibrium of a mathematical pendulum can be stabilized by fast vibration of its suspension potint [417]. In about 40 years, in the end of the 1940s Russian physicist, future Nobel prize winner Piotr Kapitsa surprised his colleagues by experiment with a rod eccentrically mounted on a horizontal motor shaft. The demonstration showed that the upper unstable equilibrium of the swinging rod (pendulum) can be made stable by sufficiently fast vibrations of the pivot. The experimental results were explained both by Kapitsa himself who developed his method of "effective potential" [224] (see also [71, 73]) and by mathematician Nikolai Bogoliubov by means of the method of averaging (history and explanations see, e.g., in [72]). The above mentioned and other results started the development of a new field in mechanics called "Vibrational mechanics" with numerous applications in science and technology [73]. Similar ideas formed the basement of a corresponding branch of the control theory: vibrational control [62, 292]. It is important to stress that Kapitsa's experiment was, perhaps, the first one clearly demonstrating the possibility and physical consequences of changing properties of a physical system by means of control.

AB - In the beginning of the 20th century, Scottish professor P. Stephenson has shown mathematically that the upper unstable equilibrium of a mathematical pendulum can be stabilized by fast vibration of its suspension potint [417]. In about 40 years, in the end of the 1940s Russian physicist, future Nobel prize winner Piotr Kapitsa surprised his colleagues by experiment with a rod eccentrically mounted on a horizontal motor shaft. The demonstration showed that the upper unstable equilibrium of the swinging rod (pendulum) can be made stable by sufficiently fast vibrations of the pivot. The experimental results were explained both by Kapitsa himself who developed his method of "effective potential" [224] (see also [71, 73]) and by mathematician Nikolai Bogoliubov by means of the method of averaging (history and explanations see, e.g., in [72]). The above mentioned and other results started the development of a new field in mechanics called "Vibrational mechanics" with numerous applications in science and technology [73]. Similar ideas formed the basement of a corresponding branch of the control theory: vibrational control [62, 292]. It is important to stress that Kapitsa's experiment was, perhaps, the first one clearly demonstrating the possibility and physical consequences of changing properties of a physical system by means of control.

UR - http://www.scopus.com/inward/record.url?scp=34347370273&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-46277-4_10

DO - 10.1007/978-3-540-46277-4_10

M3 - Chapter

AN - SCOPUS:34347370273

SN - 3540462759

SN - 9783540462750

T3 - Understanding Complex Systems

SP - 183

EP - 211

BT - Cybernetical Physics

ER -