Presented by Andrey Kuzmin Mathematical aspects of mechanical systems eigentones Department of...
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Presented by Andrey Kuzmin
Mathematical aspects of mechanical systems eigentones
Department of Applied Mathematics
Joint Advanced Student School St.Petersburg 2006
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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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1.1 History
• History of development of the linearvibration 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 centuryThe linear theory has been completed
Intro
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• Today’s problems of the linear vibration theory:
• Vibration problems of mechanical systems
1.2 Problems
– How correctly to choose degrees of freedom?
– How correctly to define external influences?
Choice of the calculated scheme
Linear statement Nonlinear statement
Intro
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• 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
1.3 Solution
allow making successive approximations
allow making any approximations
Intro
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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:*0B b 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
Rod systems
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• Construct a diagonal masses matrix M, calculate matrix
product D = BM and consider system of homogeneous equations
where
• In the end compute the determinant ,
eigenvalues and corresponding eigenvectors of matrix D
2.1 Forces Method
0 or BM E X DX X (1)
an amplitudes vector of displacements
the unit matrix
frequency of free vibrations of the given system
2
1
0BM E
Rod systems
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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
Rod systems
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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
Rod systems
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2.3 Example: the problem solution
• Calculation of displacements: evaluation of integrals on the Vereschagin's Method
• Then we construct the stiffness matrix
01 1
11
01 2
12 21
02 1
22
1,708
0,482–
0,905
l
l
l
M Mdx
EI EI
M Mdx
EI EI
M Mdx
EI EI
1,708 0,4821
0,482 0,905
BEI
Rod systems
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• The masses matrix has the form (at т1 = 2т, т2 = т):
• To find eigenvalues and eigenvectors of the matrix D = BM we compute the determinant:
2.3 Example: the problem solution
2 0
0 1
M m
3,416 0,482 0
0,964 0,905
j
j
m m
EI EIBM Em m
EI EI
Rod systems
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• Then we obtain a quadratic equation
• Thus we can find frequencies of free vibrations of the frame
• 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:
2.3 Example: the problem solution
22 4,321 2,627 0
j j
m m
EI EI
1
2
3,5891
0,7319
m
EIm
EI
1 21 2
1 10,5278 ; 1,1689
EI EI
m m
withroots
12
22
3,416 3,5891 1 0,482 0
3,416 0,7319 1 0,482 0
m m mX
EI EI EI
m m mX
EI EI EI
Rod systems
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11
5,569-0,359
2.3 Example: the problem solution
• Solving each equations separately, we find eigenvectors ν1
and ν2:
• Then we obtain forms of the free vibrations
1 2
1 1;
0,359 5,569
v v
Rod systems
Fig. 1, d. The main forms of the free vibrations
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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
Plates and shells
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3.1 Properties of eigentones
• Properties of nonlinear eigentones:– Deflections are comparable to thickness of a plate:
Rigid plates / shells Flexible plates / shells
– Frequency depends on vibration amplitude
transform
Fig. 2. Possible of dependence between
the characteristic deflection and nonlinear eigentones frequency
Plates and shells
a) Thin system b) Soft system
Skeletal line
1 1
A A
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System with infinite number degrees of freedom
System with one degree of freedom
3.2 Solution of nonlinear problems
Approximation
Plates and shells
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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:
where
24
20
D ww
h g t
(2)
The rectangular plate
3
212 1
EhD
4 4 44
4 4 2 22
x y x y
w – function of the deflection – density of the plate materialg – the free fall accelerationD – cylindrical stiffnessE – Young's modulus – the Poisson's ratio
4 – the differential functional
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3.4 Solution of the linear problem
24
20 0
sin sin 0a b D w m x n y
w dxdyh g a bt
Integration
220,2
0mn
d
dt
( )f t
h where
• Approximation of the deflection on the Kantorovich's Method:
• Substituting the equation (2) instead of function f(t):
( )sin sinm x n y
w f ta b
some temporal function
The rectangular plate
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3.4 Solution of the linear problem
Fig. 3. Character of wave
formation of the rectangular plate at
vibrations of the different form
a
b
224 4 2 2 2
220, 2 2 2 2
1
12 1mn
nm c h
m
a b
where
• The square of eigentones frequency at small deflections has form:
The rectangular plate
Egc
the velocity of spreading of longitudinal
elastic waves in a material of the plate
m = n = 1
a) the first form
m = 2, n = 1
b) the second form
m = n = 2
b) the third form
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• Examine vibrations of a plate at amplitudes which are comparable with its thickness
• Assume that the ratio of the plate sides is within the limits of
• We take advantage of the main equations of the shells theory
at kx = ky = 0:
where
3.5 The rectangular plate, fixed at edges: a nonlinear problem
a
b
1 2
24
2
4
( , )
1 1( , )
2
D ww L w
h g t
L w wE
(3)
(4)
The rectangular plate
Equilibrium equation
2 2 2 2 2 2
2 2 2 2( , ) 2
A B A B A BL A B
x y x yx y y x
differential functional
a stress function
Deformation equation
the main shell curvatures
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• Set expression (approximation) of a deflection
• Substituting (5) in the right member of the equation (4), we shall obtain the equation, which private solution is:
• Define , , where Fx and Fy – section areas
of ribs in a direction of axes x and y
3.6 Solution of the nonlinear problem
1
2 2cos cos
x yA B
a b
2 2
2
2 2
2
32
32
f aA E
b
f bB E
a
where
( )sin sinx y
w f ta b
(5)
yy
hv
Fx
x
hv
F
The rectangular plate
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• Then the solution of a homogeneous equation will have the form:
• Finally
3.6 Solution of the nonlinear problem
22 2 22 2 2cos cos
32 2 2yx
p xp yf a x b yE
b a a b
22
2 2 2yx
p xp y
2
2 22
2 2
2
2 22
2 2
1
8 1 1
1
8 1 1
y
x
x y
x
y
x y
b v
ap E fb v v
bv
ap E fb v v
where
the stresses applied to the plate through boundary ribs (they are considered as positive at a tensioning)
4 0
The rectangular plate
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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
• Generally we approximate functions w(x,y,t) in the form of series
24
2( , )
D wX w L w
h g t
1
n
i ii
w f
some given and independent functions which satisfy to boundary conditions of a problem
the parameters depending on t
The rectangular plate
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3.7 Solution: the first stage of approximation
• On the Bubnov-Galerkin’s Method we write out n equations of type
• In our solution η1 has the form
0, 1,2,...,i
F
X dxdy i n (6)
1 sin sinx y
a b
The rectangular plate
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• Hence, integrating (6) and passing to dimensionless parameters, we obtain the equation
where the dimensionless parameters K and ζ have the form
3.7 Solution: the first stage of approximation
2
2 202
1 0d
Kdt
(7)
2 2
2 2 2 2 2 42
2 2
1,5 1 0.75 11 1 11 1
1 11 1 1 1
yx
x y
vK v
v v
(8)
( )f t
h
The rectangular plate
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• Hence, integrating (6) and passing to dimensionless parameters, we obtain the equation
• Parameter – the square of the main frequency of the plate eigentones:
3.7 Solution: the first stage of approximation
2
2 202
1 0d
Kdt
(7)
20
24 2 22 20 2 2
1
12 1
hc
ab
The rectangular plate
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3.7 Solution: the first stage of approximation
24
2( , )
D ww L w
h g t
2
2 202
1 0d
Kdt
Bubnov-Galerkin’s Method
– the nonlinear differential partial equation of the fourth degree
– the nonlinear differential equation in ordinary derivatives of the second degree
2 stage1 stage
= ?
Integration
The rectangular plate
• Thus
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• Consider the simply supported plate
• Let's present temporal function in the form
3.8 Solution: the second stage of approximation
0
0
0
x
y
x y
Fx
F
y
p p
v
v
from (8) hence
2 4
22
3 1 1
4 1K
The rectangular plate
cosA t
vibration frequency
dimensionless amplitude
(9)
that is ribs are absent
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• Let
• Further integrate Z over period of vibrations
• We obtain the equation expressing dependence between
frequency of nonlinear vibrations ω and amplitude A:
3.8 Solution: the second stage of approximation
2T
2 /
0
( ) cos( ) 0Z t t dt
2 2 20
31
4KA
The rectangular plate
2
2 202
( ) 1d
Z t Kdt
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• Define
• Then
3.8 Solution: the second stage of approximation
0
frequency of nonlinear vibrations
frequency of linear vibrations
2 231
4KA
Fig. 4. A skeletal line of the thin type for ideal
rectangular plate at nonlinear vibrations of the general form
The rectangular plate
A
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• Now we consider shallow and rectangular
in a plane of the shell
• The main shell curvatures kx, ky are assumed by constants:
3.9 The bicurved shell
1 2
1 1x yk k
R R
Fig. 5. The shallow
bicurved shell.
The bicurved shell
where R1,2 – radiuses of
curvature
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• The dynamic equations of the nonlinear theory of shallow shells have the form:
where the differential functional
• For full and initial deflections are define by
3.10 The bicurved shell: the problem setup
24 2
0 2
4 20 0 0
( , )
1 1( , ) ( , )
2
k
k
D ww w L w
h t
L w w L w w w wE
2 22
2 2k x y
A AA k k
y x
The bicurved shell
0 0( )sin sin sin sinx y x y
w f t w fa b a b
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• Using the method considered above, we obtain the following ordinary differential equation of shell vibrations
• Here
The square of the main frequency of ideal shell eigentones at small deflections has the form
3.11 The bicurved shell: the problem solution
2
2 2 302
0d
dt
(10)
010 1 0
( )
ff tf f f
h h
2 2 220 2 2
c h
a b
22 2 2
2*22 2 2 2
1
12 1 1k
The bicurved shell
where
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Here variables , , have the form
3.11 The bicurved shell: the problem solution
2
2 2 302
0d
dt
(10)
4 * *2 44
02 4 2 4 22 2
16 162 8 81 1 1 1
312 1 1
y xk k
4 * *2 44
02 4 2 4 22 2
16 161 8 1 8 91
2 2 412 1 1
y xk k
2
42
0,75 112
The bicurved shell
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3.11 The bicurved shell: the problem solution
• Thus we obtain the following equation for definition of an amplitude-frequency characteristic
2 28 31
3 4A A
where0
The bicurved shell
Fig. 6. The amplitude-frequency dependences for shallow
shells of various curvature
A
2
4
6
8
0 1 2
shell at
cylindrical shell at
plate at
* * 24x yk k
* * 0x yk k
* *0, 24x yk k
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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.