Two Classical Problems on Stabilization of Statically Unstable Systems by Vibration Alexander P. Seyranian and Аndrei А. Seyranian MSU n.a. Lomonosov MSTU n.a. Bauman

- small parameters Non-dimensional variables Hill’s equation with damping Stephenson (1908) Kapitza (1951) Stabilization condition What is new: damping and arbitrary periodic function Stabilization of the inverted pendulum by HF excitation

Instability regions for Mathieu-Hill equation with damping Comparison between analytical and numerical results

Instability regions for the case Destabilization effect of small damping

Stabilization frequency for the pendulum General formula for symmetric functions

Stabilization frequency for the pendulum For symmetric function For non-symmetric function

Stabilization frequency for the pendulum For piecewise constant function we have The first term of this formula can be compared with the formulas derived in the famous books by V.I.Arnold: is important ! Ordinary Differential Equations, The MIT Press, Mathematical Methods in Classical Mechanics, Springer, 1989.

Stabilization of straight position of elastic column under axial periodic force exceeding critical (Euler) value  Transverse vibrations of the column: (1)  The Chelomei problem (1956)

Reduction to ordinary differential equations  Simply supported ends  Separation of variables  New notation 

V.N. Chelomei: «high frequency» stabilization of the column  Assumption:, similarity with stabilization of the inverted pendulum  Equation (2): perturbation method, averaging method,.   Contradictions: Critical excitation frequency is of the order of the main eigenfrequency Excitation frequency is not limited from below  V.N.Chelomei ( )  V.N.Chelomei. On increasing of stability prorerties of elastic systems by vibration. Doklady AN SSSR V N 3. P  V.N.Chelomei. Paradoxes in mechanics caused by vibration. Doklady AN SSSR V N 1. P

Short review of previous research  N.N.Bogolyubov, Yu.A.Mitroplolskii Asymptotic methods in the theory of nonlinear vibrations. Moscow, Nauka, p.  V.V.Bolotin Numerical analysis Similarity with the problem on stabilization of an inverted pendulum does not take place due to interference of resonance regions of higher harmonics, narrowing stabilization region of the column Vibrations in Engineering. Handbook. V. 1. Vibrations of linear systems. Moscow: Mashinostroenie, p.  Jensen J.S., Tcherniak D.M., Thomsen J.J. Under high frequency excitation the straight equilibrium position exists along with the curved stable position Effect of increase of stiffness (eigenfrequencies of transverse vibrations) under high frequency excitation is confirmed experimentally, but critical stability forces or frequencies were not studied

Analysis of Stabilization Region of the Column  Obtaining upper boundary for stabilization frequency: We apply the results for stability regions study for Hill’s equation with damping to equation (2) at assuming that  Seyranian A.P. Resonance regions for Hill’s equation with damping // Doklady AN V N 1. P  Seyranian A.A., Seyranian A.P. On stability of an inverted pendulum with vibrating suspension point // J. Appl. Maths. Mechs V. 70. N 5. P  Upper boundary: (4)

Analysis of Stabilization Region of the Column  Obtaining lower boundary of stabilization frequency: Strutt-Ince diagram  Analysis of stability region near first critical frequency  Lower boundary: (5)

Stabilization Region  Independent parameters and  Damping decreases upper as well as lower critical frequency  Stabilization region exists only at rather high excitation amplitude (6)

Numerical Results  Good agreement between analytical and numerical results

Stabilization of the column at given excitation frequency    

Influence of instability regions of equation (2) for higher harmonics  Parametric resonance for Mathieu-Hill equations (2) occurs at frequencies close to the values [8, 10]:  :   Numerical results confirm this conclusion

Conclusions  Stability regions for Hill’s equation with small damping and arbitrary periodic excitation function near zero frequency are obtained  Formulae for the critical stabilization frequencies of the inverted pendulum are derived  Destabilization effect of small damping is recognized  Unlike the inverted pendulum an elastic column is stabilized by frequencies of the order of the main eigenfrequency of transverse vibrations belonging to some interval  It is shown that instability regions for higher harmonics k=2,3,…do not influence the stabilization region  Numerical results confirm validity and accuracy of the obtained analytical formulae

References 1. В.Н.Челомей. О возможности повышения устойчивости упругих систем при помощи вибрации. Доклады АН СССР Т № 3. С В.Н.Челомей. Парадоксы в механике, вызываемые вибрациями. Доклады АН СССР Т № 1. С В.Н.Челомей. Избранные труды. М.: Машиностроение, с. 4. Боголюбов Н.Н., Митропольский Ю.А. Асимптотические методы в теории нелинейных колебаний. М: Наука, с. 5. Вибрации в технике. Справочник. Т. 1. Колебания линейных систем / Под ред. В.В. Болотина. М.: Машиностроение, с. 6. Jensen J.S. Buckling of an elastic beam with added high-frequency excitation // International Journal of Non-Linear Mechanics V.35. P Jensen J.S., Tcherniak D.M., Thomsen J.J. Stiffening effects of high- frequency excitation: experiments for an axially loaded beam // ASME Journal of Applied Mechanics V. 67. P Сейранян А.П. Области резонанса для уравнения Хилла с демпфированием // Доклады АН Т № 1. С Сейранян А.А., Сейранян А.П. Об устойчивости перевернутого маятника с вибрирующей точкой подвеса // Прикладная математика и механика Т. 70. № 5. С Меркин Д.Р. Введение в теорию устойчивости движения. М.: Наука, с. 11. Пановко Я.Г., Губанова И.И. Устойчивость и колебания упругих систем. М.: Наука, Thomsen J.J. Vibrations and Stability. Advanced Theory, Analysis and Tools. Berlin: Springer, p.

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