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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.
Shnorhakalutyun!
Спасибо за внимание!
Au revoir!
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