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

0sin)( zgmrcI )( taz

0)]([ 2

- small parameters

Non-dimensional variables

I

cg

a 20

0

Hill’s equation with dampingStephenson (1908)Kapitza (1951)

32

7

22

42222 Stabilization

condition cos)(

What is new: damping and arbitrary periodic function

Stabilization of the inverted pendulum by HF excitation

Instability regions for Mathieu-Hill equation with damping ,0)]([ cos

Comparison betweenanalytical and numerical results

Instability regions for the case 3cos

Destabilization effect of small damping

Stabilization frequency for the pendulum

General formula for symmetric functions

3

2

220

0 28

3

22

1

FF

H

FF

L

F

K

FF

L

F

)()(

0)()(1

)(2

1 2

0 0

22

0

dtdtdtttFt

Stabilization frequency for the pendulum

18432

2389

432

712

320

0

For symmetric function cos)(

For non-symmetric function 2

1

4

1)(

3

320

2

0

028.0214.0045.0162.0202.019.2

Stabilization frequency for the pendulum

For piecewise constant

function we have

1, 0 ( ) =

1, 2

3 3 2 7 30

0

2 793

126 120 970200

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, 1978.

Mathematical Methods in Classical Mechanics, Springer, 1989.

2

Stabilization of straight position of elastic column under axial periodic force exceeding critical (Euler) value

Transverse vibrations of the column:

(1)

022

2

2

2

4

4

t

um

t

um

x

utP

x

uEJ

tPPtP t 0

x - coordinate along axis of column, t -time, txu , - deflection of column, m - mass

per unit length, EJ - flexural stiffness, - damping coefficient, tP , - amplitude and excitation frequency of axial vibration

The Chelomei problem (1956)

Reduction to ordinary differential equations

Simply supported ends

Separation of variables

New notation

0),(),0( tlutu2 2

2 2

0

0x x l

u u

x x

, sin ( / )jj

u x t t j x l kk / , .t

012 0

2

2

2

k

k

t

k

kkkk

k

P

P

P

P

d

d

d

d

, ,...2,1k (2)

222 // lmEJkk ─ k -th eigenfrequency of transverse vibration, 222 / lEJkPk ─ k -th critical (Euler) force

Trivial solution 0),( txu is asymptotically stable, if every 0)( tk

while t , ,...2,1k ., and unstable, if at least one of )(tk becomes unbounded while t

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 (1914-1984)

V.N.Chelomei. On increasing of stability prorerties of elastic systems by vibration. Doklady AN SSSR. 1956. V. 110. N 3. P. 345-347.

V.N.Chelomei. Paradoxes in mechanics caused by vibration. Doklady AN SSSR.

1983. V. 270. N 1. P. 62-67.

1

1k

21

22

1

42

(3)

1/ PPt , 1/ 10 PP , 11 / .

Stabilization region for the column

( cos )

Short review of previous research

N.N.Bogolyubov, Yu.A.Mitroplolskii

Asymptotic methods in the theory of nonlinear vibrations. Moscow, Nauka, 1974. 503 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, 1999. 504 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

HTML Document

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. 2001. V. 376. N 1. P. 44-47. Seyranian A.A., Seyranian A.P. On stability of an inverted pendulum

with vibrating suspension point // J. Appl. Maths. Mechs. 2006.

V. 70. N 5. P. 835-843.

Upper boundary:

1k

11/0 10 PP 1/ 1 PPt cos

2 22

11

74

2 8

(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: 2

1

21

22

222

(5)

22

2

222

22

22

2

JPEG Image

Stabilization Region

Independent parameters and

Damping decreases upper as well as

lower critical frequency Stabilization region exists only at rather

high excitation amplitude

21

22

1

21

22 4

87

2222

(6)

05.01

, 1

Numerical Results

Good agreement between analytical and numerical results

05.01 05.01

Stabilization of the column at given excitation frequency

2/2

1/ 1

05.01

Approximate formula for stabilization region when and 1 are small

At moderate amplitude of excitation the column can be stabilized when constant part of the axial force is only slightly higher than Euler’s value, 11/ 10 PP

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

,3,2k

41

20

2n

P

P

k

k

, ,2,1n

21

02

1 12

kP

P

n

k

, ,2,1n

2k 134 , 132 , 13/34 , ,3 1

3k1212 , 126 , 124 , ,23 1

JPEG Image

When system is damped the regions of instability for equation (2) with high ,3,2k do not influence stabilization region (6)

For Mathieu-Hill equation (2) only first instability regions which start from the points

134 and 1212 are wide, and at moderate amplitudes of excitation even with small damping the instability regions corresponding to big n disappear

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

References1. В.Н.Челомей. О возможности повышения устойчивости упругих систем при помощи вибрации.

Доклады АН СССР. 1956. Т. 110. № 3. С. 345-347.2. В.Н.Челомей. Парадоксы в механике, вызываемые вибрациями. Доклады АН СССР. 1983. Т. 270.

№ 1. С. 62-67.3. В.Н.Челомей. Избранные труды. М.: Машиностроение, 1989. 335 с.4. Боголюбов Н.Н., Митропольский Ю.А. Асимптотические методы в теории нелинейных

колебаний. М: Наука, 1974. 503 с.5. Вибрации в технике. Справочник. Т. 1. Колебания линейных систем / Под ред. В.В. Болотина.

М.: Машиностроение, 1999. 504 с.6. Jensen J.S. Buckling of an elastic beam with added high-frequency excitation // International

Journal of Non-Linear Mechanics. 2000. V.35. P. 217-227.7. 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. 2000. V. 67. P. 397-402.

8. Сейранян А.П. Области резонанса для уравнения Хилла с демпфированием // Доклады АН. 2001. Т. 376. № 1. С. 44-47.

9. Сейранян А.А., Сейранян А.П. Об устойчивости перевернутого маятника с вибрирующей точкой подвеса // Прикладная математика и механика. 2006. Т. 70. № 5. С. 835-843.

10. Меркин Д.Р. Введение в теорию устойчивости движения. М.: Наука, 1987. 304 с.11. Пановко Я.Г., Губанова И.И. Устойчивость и колебания упругих систем. М.: Наука, 1987.12. Thomsen J.J. Vibrations and Stability. Advanced Theory, Analysis and Tools. Berlin: Springer,

2003. 404 p.

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Спасибо за внимание!

Au revoir!

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