Задача В.И. Челомея

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
Stabilization of the inverted pendulum by HF excitation
What is new:
Non-dimensional variables
damping and
arbitrary periodic
function
c

I
z  a (t )
-
a02

g
0


small parameters
I c  mr ( g  z) sin   0 Hill’s equation with damping
Stephenson (1908)
     [ 2  ( )]   0

Kapitza (1951)
 ( )  cos
Stabilization
condition
 2  
2
2

 2 2
7 4

2
32
Instability regions for Mathieu-Hill equation
with damping      [  ( )]   0 ,   cos
Comparison between
analytical and
numerical results
Instability regions for the case   cos 
3
Destabilization
effect of small
damping
Stabilization frequency for the pendulum
General formula for symmetric functions  (   )   ( )
 3
K  02  3 L2

1
L
H




  2

 0   F 2 F  F 2  F  8F  F 2 F  F 
2
2
t
 1
 1 2
F  
t (t )dt     (t )  ( ) d dt  0

0
 2 0
  0
Stabilization frequency for the pendulum
For symmetric function  ( )  cos
 1 7  

2389  
 2 



0

32
4
18432


2
0
3
1  1

(

)

For non-symmetric function
  
4  2
3
 2.19

 0.202  0.162   0.045  2  0.214   02  0.028  3
0

Stabilization frequency for the pendulum
For piecewise constant
function we have
 1, 0    
 ( ) = 
1,     2
3
3
2
7 3


2      0 79  
 3




0
120
970200 
   126
The first term of this formula can be compared with the formulas
derived in the famous books by V.I.Arnold:
2 is important !
Ordinary Differential Equations, The MIT Press, 1978.
Mathematical Methods in Classical Mechanics, Springer, 1989.
Stabilization of straight position of elastic column under
axial periodic force exceeding critical (Euler) value
The Chelomei
problem (1956)

Transverse vibrations of the column:
 4u
 2u
u
 2u
EJ 4  Pt  2  2 m  m 2  0
x
x
t
t


(1)
x - coordinate along axis of column, t -time, ux, t  - deflection of column, m - mass
per unit length, EJ - flexural stiffness,  - damping coefficient, Pt ,  - amplitude
and excitation frequency of axial vibration
Pt   P0  Pt   t 
Reduction to ordinary differential equations


Simply supported ends
Separation of variables
u (0, t )  u (l , t )  0
New notation
x 0
u  x, t     j  t  sin ( j x / l )
j

 2u
 x2
 k   /  k ,    t.
 2u
 2
x
0
x l

k   2 k 2 EJ / m / l 2 ─ k -th eigenfrequency of transverse vibration,
Pk   2 k 2 EJ / l 2 ─ k -th critical (Euler) force


d k
  k  d k   k 

2




k
2
d
   d   
2
2
 P0 Pt  

1 
 k  0 , k  1, 2,... (2)
Pk 
 Pk
Trivial solution u ( x, t )  0 is asymptotically stable, if every  k (t )  0
while t   , k  1, 2,... ., and unstable, if at least one of  k (t ) becomes
unbounded while t  
V.N. Chelomei: «high frequency» stabilization of
the column



Assumption:   1,
similarity with stabilization of the inverted
pendulum
Equation (2): perturbation method,
averaging method, k  1 .
Stabilization region for the column
(     cos )
2
 
2
  
 412
2
 1 
(3)
  Pt / P1 ,   P0 / P1  1 , 1   / 1 .

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.
Short review of previous research

N.N.Bogolyubov, Yu.A.Mitroplolskii


V.V.Bolotin




Asymptotic methods in the theory of nonlinear vibrations. Moscow, Nauka, 1974. 503 p.
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
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 k  1 assuming that
    cos 
0    P0 / P1  1  1   Pt / P1  1



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:
2
 
2
7
2


4


 
1
2
8
 1 
(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 
  2 2

     2 


 
2

   1   
2
 1 
2
  2 2
(5)
2
 

2 

2
2
2
  2 
2

2
2
Stabilization Region
  2 
  2 
2
2
 
 2 7

       

 412
2
2 8
 1 

2
2
1

Independent parameters  ,  and 1

Damping decreases upper as well as
lower critical frequency
Stabilization region exists only at rather
high excitation amplitude

1  0.05
(6)
Numerical Results
1  0.05

1  0.05
Good agreement between analytical and numerical results
Stabilization of the column at given excitation
frequency
 / 1  1
1  0.05

Approximate formula for stabilization region when  and 1 are small
 2 /2

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,   P0 / P1  1  1
Influence of instability regions of equation (2)
for higher harmonics k  2,3,

Parametric resonance for Mathieu-Hill equations (2) occurs at frequencies close to
the values [8, 10]:
2
 P0  n 2
2



k
P0
21k

 1   
, n  1,2,


1


n

1
,
2
,

,

P
4
2

 
k 
n
P1 k

k 2 :
4 3  1 , 2 3  1 , 4 3 / 3 1 ,

k 3 :
12 2 1 , 6 2 1 , 4 2 1 , 3 2 1 ,



3 1 , 
For Mathieu-Hill equation (2) only first instability regions which start from the points
4 3  1 and 12 2 1 are wide, and at moderate amplitudes of excitation even with
small damping the instability regions corresponding to big n disappear
When system is damped the regions of instability for equation (2) with high
k  2,3, do not influence stabilization region (6)
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.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
В.Н.Челомей. О возможности повышения устойчивости упругих систем при помощи вибрации.
Доклады АН СССР. 1956. Т. 110. № 3. С. 345-347.
В.Н.Челомей. Парадоксы в механике, вызываемые вибрациями. Доклады АН СССР. 1983. Т. 270.
№ 1. С. 62-67.
В.Н.Челомей. Избранные труды. М.: Машиностроение, 1989. 335 с.
Боголюбов Н.Н., Митропольский Ю.А. Асимптотические методы в теории нелинейных
колебаний. М: Наука, 1974. 503 с.
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Машиностроение, 1999. 504 с.
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.
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.
Сейранян А.П. Области резонанса для уравнения Хилла с демпфированием // Доклады АН. 2001.
Т. 376. № 1. С. 44-47.
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.
Меркин Д.Р. Введение в теорию устойчивости движения. М.: Наука, 1987. 304 с.
Пановко Я.Г., Губанова И.И. Устойчивость и колебания упругих систем. М.: Наука, 1987.
Thomsen J.J. Vibrations and Stability. Advanced Theory, Analysis and Tools. Berlin: Springer,
2003. 404 p.

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
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
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
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