Department of Applied Mathematics
Mathematical aspects of
mechanical systems eigentones
Presented by Andrey Kuzmin
Agenda
PART I.
Introduction to the theory of mechanical vibrations
PART II.
Eigentones (free vibrations) of rod systems
– Forces Method
– Example
PART III.
Eigentones of plates and shells
– Properties of eigentones
– The rectangular plate: linear and nonlinear statement
– The bicurved shell
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2
PART I
Introduction to the theory of
mechanical vibrations
Intro
1.1 History
• History of development of the linear
vibration theory:
– XVIII century
“Analytical mechanics” by Lagrange –
systems with several degrees of freedom
– XIX century
Rayleigh and others – systems with the infinite
number degrees of freedom
– XX century
The linear theory has been completed
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Intro
1.2 Problems
• Today’s problems of the linear vibration theory:
– How correctly to choose
degrees of freedom?
– How correctly to define
external influences?
Choice of the
calculated
scheme
• Vibration problems of mechanical systems
Linear statement
Nonlinear statement
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Intro
1.3 Solution
• Role of the nonlinear theory:
The phenomena description escaping from a
field of vision at any attempt to linearize the
considered problem.
• Approximate solution methods of nonlinear
problems:
–
–
–
–
Poincare and Lyapunov’s Methods
Krylov-Bogolyubov's Method
Bubnov-Galerkin’s Method
and others
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allow making
successive
approximations
allow making any
approximations
6
PART II
Eigentones (free vibrations) of
rod systems
Rod systems
2.1 Forces Method
•
•
•
Consider rod systems in which the distributed mass is
concentrated in separate sections (systems with a finite
number of degrees of freedom)
Define displacements from a unit forces applied in directions
of masses vibrations
Construct the stiffness matrix of system:
B  b0* fb
the gain matrix depend on the unit forces applied
in a direction of masses vibrations in the given
system
the stiffness matrix of separate elements
transposition of the matrix equal to the matrix b,
constructed for statically definable system
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Rod systems
2.1 Forces Method
•
Construct a diagonal masses matrix M, calculate matrix
product D = BM and consider system of homogeneous
equations
 BM   E  X  0 or
where

1
2
DX   X
an amplitudes vector of displacements
the unit matrix
frequency of free vibrations
given system
•
(1)
of the
In the end compute the determinant BM   E  0,
eigenvalues and corresponding eigenvectors of matrix D
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Rod systems
2.2 Example: the problem setup
• Define frequencies and forms of the free vibrations of a
statically indeterminate frame with two concentrated masses
т1 = 2т, т2 = т and identical stiffnesses of rods at a
bending down (EI = const, where E – Young's modulus; I –
Inertia moment of section)
Fig. 1, a.
Rod system with two
degree of freedoms
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Rod systems
2.3 Example: the problem solution
Fig 1, b.
The bending moments
stress diagrams depend
on the unit forces applied
in a direction of masses
vibrations
Fig 1, c.
The stress diagrams
depend on the same unit
forces in statically
determinate system
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Rod systems
2.3 Example: the problem solution
• Calculation of displacements: evaluation of integrals on the
Vereschagin's Method
M 1 M 10
1, 708
11   
dx 
EI
EI
l
12   21
 22
M 1 M 20
0, 482
 
dx  –
EI
EI
l
M 2 M 10
0,905
 
dx 
EI
EI
l
• Then we construct the stiffness matrix
B
1
EI
 1, 708
 0, 482

 0, 482 
0,905 
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Rod systems
2.3 Example: the problem solution
• The masses matrix has the form (at т1 = 2т, т2 = т):
2 0
M  m

0 1 
• To find eigenvalues and eigenvectors of the matrix D = BM
we compute the determinant:
m
 j
EI
BM   E 
m
0,964
EI
3, 416
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m
EI
0
m
0,905
 j
EI
 0, 482
13
Rod systems
2.3 Example: the problem solution
• Then we obtain a quadratic equation
m



3,5891
1
m
EI
 m
with 
2
 j  4,321  j  2, 627    0

roots 
m
EI
 EI 
2  0, 7319

EI
Thus we can find frequencies of free vibrations of the frame
2
•
1 
1
1
 0,5278
EI
;
m
2 
1
2
 1,1689
EI
m
• For definition of corresponding forms of vibrations we use (1).
Let, for example, X1 = 1. From the first equation we find Х2 for
each value of λj: 
m
m
m  1

3,
416

3,5891

1

0,
482


 X2  0

EI
EI 
EI 



 3, 416 m  0, 7319 m   1   0, 482 m  X  2  0


 2

EI
EI 
EI 

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Rod systems
2.3 Example: the problem solution
•
Solving each equations separately, we find eigenvectors ν1
and ν2:
 1

 1

v1  
;
v

2

 5,569 
 0,359 


•
Then we obtain forms of the free vibrations
1
Fig. 1, d.
-0,359
1
5,569
The main forms of
the free vibrations
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PART III
Eigentones of plates and shells
Plates and shells
3.1 Properties of eigentones
• Properties of linear eigentones (free vibrations):
– Plates and shells – systems with infinite number
degrees of freedom. That is:
• number of eigenfrequencies is infinite
• each frequency corresponds a certain form of vibrations
– Amplitudes do not depend on frequency and are
determined by initial conditions:
• deviations of elements of a plate or a shell from equilibrium
position
• velocities of these elements in an initial instant
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Plates and shells
3.1 Properties of eigentones
• Properties of nonlinear eigentones:
– Deflections are comparable to thickness of a plate:
transform
Rigid plates / shells
Flexible plates / shells
– Frequency depends on vibration amplitude
A
A
Fig. 2.
Possible of dependence between
the characteristic deflection and
nonlinear eigentones frequency
Skeletal line
1

a) Thin system
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
1
b) Soft system
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Plates and shells
3.2 Solution of nonlinear problems
System with infinite number degrees of freedom
Approximation
System with one degree of freedom
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The rectangular plate
3.3 The rectangular plate, fixed at
edges: a linear problem
•
Let a, b – the sides of a plate
h – the thickness of a plate
•
Linear equation for a plate:
D 4
 2 w
 w
0
2
h
g t
(2)
w – function of the deflection
where
D
Eh
3
12 1   2 
4
4
4
  4  4 2 2 2
x
y
x y
4
 – density of the plate material
g – the free fall acceleration
D – cylindrical stiffness
E – Young's modulus
 – the Poisson's ratio
4 – the differential functional
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The rectangular plate
3.4 Solution of the linear problem
•
Approximation of the deflection on the Kantorovich's Method:
w  f (t ) sin
m x
n y
sin
a
b
some temporal function
•
Substituting the equation (2) instead of function f(t):
D 4
  2 w  m x n y
0 0  h  w  g t 2  sin a sin b dxdy  0
a b
Integration
d 2
2


0, mn   0
2
dt
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f (t )
where  
h
21
The rectangular plate
3.4 Solution of the linear problem
•
The square of eigentones frequency at small deflections has
2
form:
2

n 2 2 2
 m 1  2   c h
m



12 2 1   2  a 2 b 2
4

2
0, mn
4
a
where   b
the velocity of spreading of longitudinal
elastic waves in a material of the plate
c
Eg

Fig. 3.
Character of wave
formation of the
rectangular plate at
vibrations of the
different form
m=n=1
m = 2, n = 1
a) the first form b) the second
form
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m= n=2
b) the third
form
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The rectangular plate
3.5 The rectangular plate, fixed at
edges: a nonlinear problem
• Examine vibrations of a plate at amplitudes which are
comparable with its thickness
a


• Assume that the ratio of the plate sides
is within the
b
limits of 1    2
• We take advantage of the main equations of the shells theory
at kx = ky = 0:
a stress function
the main shell
curvatures
where
D 4
 2 w
 w  L( w, ) 
h
g t 2
1 4
1
    L( w, w)
E
2
Equilibrium
equation
(3)
Deformation
equation
(4)
2 A 2 B 2 A 2 B
2 A 2 B
L( A, B)  2
 2
2
2
2
xy xy
x y
y x
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differential
functional
23
The rectangular plate
3.6 Solution of the nonlinear
problem
• Set expression (approximation) of a deflection
w  f (t ) sin
x
a
sin
y
(5)
b
• Substituting (5) in the right member of the equation (4), we
shall obtain the equation, which private solution is:

 A  E

B  E

f2
32
f2
32
a2
b2
b2
a2
2 x
2 y
where
 B cos
a
b
h
h
 v y , where Fx and Fy – section areas
 vx ,
• Define
Fy
Fx
1  A cos
of ribs in a direction of axes x and y
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The rectangular plate
3.6 Solution of the nonlinear problem
• Then the solution of a homogeneous equation  4   0 will
have the form:
2
px y 2 p y x
where
2 

2
2
the stresses applied to the plate
through boundary ribs (they are
considered as positive at a tensioning)

b 2 1  v y 


2

2
a2
p  E
f
x
2

8b 1  vx  1  v y    2

b2

 2  1  vx

2
2
a
f
 py  E 2
8b 1  vx  1  v y    2

• Finally
2
2
2
2


p
x
px y
f a
2 x  b 
2 y
y
E
cos

cos


 

 
32  b 
a
b 
2
2
a
2
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The rectangular plate
3.7 Solution: the first stage of
approximation
• Apply the Bubnov-Galerkin’s Method to the equation (3) for
some fixed instant t
• Suppose X has the form
D 4
 2 w
X   w  L(w, ) 
h
g t 2
• Generally we approximate functions w(x,y,t) in the form of
series
the parameters depending on t
n
w   fii
i 1
some given and independent functions which
satisfy to boundary conditions of a problem
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The rectangular plate
3.7 Solution: the first stage of
approximation
• On the Bubnov-Galerkin’s Method we write out n equations of
type
 X dxdy  0,
i  1, 2,..., n
i
(6)
F
• In our solution η1 has the form
1  sin
x
a
sin
y
b
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The rectangular plate
3.7 Solution: the first stage of
approximation
• Hence, integrating (6) and passing to dimensionless
parameters, we obtain the equation
d 2
2
2


1

K

 0

0 
2
dt
(7)
where the dimensionless parameters K and ζ have the form
1,5 1   2 

1  vy

K

1

v



 2

x
2
2


1 



2 
1  2  1  vx  1  v y    
  
 
2
 1  0.75 1    
1 

1

 2
2 
4 
1    

  
1  2 
  
f (t )
h
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(8)
The rectangular plate
3.7 Solution: the first stage of
approximation
• Hence, integrating (6) and passing to dimensionless
parameters, we obtain the equation
d 2
2
2


1

K

 0

0 
2
dt
(7)
• Parameter 0 – the square of the main frequency of the plate
eigentones:
2
 1  
4
02 

2 2
2  h 
c
 
12 2 1   2   ab 
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The rectangular plate
3.7 Solution: the first stage of
approximation
•
Thus
D 4
 2 w
– the nonlinear
 w  L( w, ) 
2
h
g t
differential partial
1 stage
Bubnov-Galerkin’s
Method
equation of the fourth
degree
d 2
2
2


1

K

  0 – the nonlinear


0
2
dt
differential equation
2 stage
Integration
in ordinary
derivatives of the
second degree
=?
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The rectangular plate
3.8 Solution: the second stage of
approximation
• Consider the simply supported plate
px  p y  0
that is ribs
are absent


Fx  0
vx 



Fy  0
  
v y 
from (8) hence
K
3 1   2 1   4 
4 1  

2 2
• Let's present temporal function in the form
  A cos t
(9)
vibration frequency
dimensionless amplitude
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The rectangular plate
3.8 Solution: the second stage of
approximation
• Let
d 2
Z (t )  2  02 1  K  2  
dt
• Further integrate Z over period of vibrations T 
2 / 

2

Z (t ) cos( t ) dt  0
0
• We obtain the equation expressing dependence between
frequency of nonlinear vibrations ω and amplitude A:
 3

   1  KA2 
 4

2
2
0
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The rectangular plate
3.8 Solution: the second stage of
approximation
• Define
• Then


0
frequency of nonlinear vibrations
frequency of linear vibrations
A
3
  1  KA2
4
2
Fig. 4.
A skeletal line of the thin type for ideal
rectangular plate at nonlinear vibrations of
the general form

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The bicurved shell
3.9 The bicurved shell
• Now we consider shallow and rectangular
in a plane of the shell
Fig. 5.
The shallow
bicurved shell.
• The main shell curvatures kx, ky are assumed by constants:
1
kx 
R1
1
ky 
R2
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where R1,2 – radiuses
of curvature
34
The bicurved shell
3.10 The bicurved shell: the
problem setup
• The dynamic equations of the nonlinear theory of shallow
shells have the form:
2
D 4

w
2
  w  w0   L( w, )   k    2
h
t
1 4
1
     L( w, w)  L( w0 , w0 )    2k  w  w0 
E
2
2 A
2 A
where the differential functional  A  k x 2  k y 2
y
x
For full and initial deflections are define by
2
k
•
w  f (t ) sin
x
a
sin
y
b
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w0  f 0 sin
x
a
sin
y
b
35
The bicurved shell
3.11 The bicurved shell: the
problem solution
• Using the method considered above, we obtain the following
ordinary differential equation of shell vibrations
d 2
2
2
3







0

0 
2
dt
• Here
 
f1 (t )
h
0 
f0
h
(10)
f1  f  f0
The square of the main frequency of ideal shell eigentones at
small deflections has the form
 
2
0
 ch
2 2
a 2b2
2
 where
 1  
2

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
2 2
12 2 1   2 

2
 2 1   2 
2
k* 
36
2
The bicurved shell
3.11 The bicurved shell: the
problem solution
d 2
2
2
3







0


0
2
dt
(10)
Here variables , ,  have the form



4
8

1 


 1   2 2  





4 * 


* 
4
16
k
 2 16 k y  1
8
1
8

9

4
x 







1






0
12 2    4  2 1   2 2   4  2 1   2 2  4









*
16

k
16
k
 2 2
8
y
4
x
1 

  1
 1     0 
2
4
2
4
 1   2   
12   3



4
*
2
4
  0, 75
1




12 2 
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The bicurved shell
3.11 The bicurved shell: the
problem solution
• Thus we obtain the following equation for definition of an
amplitude-frequency characteristic
3 2
  1
A
A
3
4
8
2
where


0
A
shell at kx*  k y*  24
8
*
*
cylindrical shell at kx  0, k y  24
6
Fig. 6.
4
The amplitude-frequency
dependences for shallow
shells of various curvature
2
plate at kx*  k *y  0
0
1
Joint Advanced Student School
St.Petersburg 2006
2

38
Applications
Joint Advanced Student School
St.Petersburg 2006
39
References
• Ilyin V.P., Karpov V.V., Maslennikov A.M.
Numerical methods of a problems solution of building mechanics. –
Moscow: ASV; St. Petersburg.: SPSUACE, 2005.
• Karpov V.V., Ignatyev O.V., Salnikov A.Y.
Nonlinear mathematical models of shells deformation of variable
thickness and algorithms of their research. – Moscow: ASV; St.
Petersburg.: SPSUACE, 2002.
• Panovko J.G., Gubanova I.I.
Stability and vibrations of elastic systems. – Moscow: Nauka. 1987.
• Volmir A.S.
Nonlinear dynamics of plates and shells. – Moscow: Nauka. 1972.
Joint Advanced Student School
St.Petersburg 2006
40